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<div class="section" id="fwk-redden-ch07_s05" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">7.5</span> Solving Exponential and Logarithmic Equations</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch07_s05_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch07_s05_o01" numeration="arabic">
<li>Solve exponential equations.</li>
<li>Use the change of base formula to approximate logarithms.</li>
<li>Solve logarithmic equations.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch07_s05_s01" version="5.0" lang="en">
<h2 class="title editable block">Solving Exponential Equations</h2>
<p class="para block" id="fwk-redden-ch07_s05_s01_p01">An <span class="margin_term"><a class="glossterm">exponential equation</a><span class="glossdef">An equation which includes a variable as an exponent.</span></span> is an equation that includes a variable as one of its exponents. In this section we describe two methods for solving exponential equations. First, recall that exponential functions defined by <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1353" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>b</mi><mi>x</mi></msup></mrow></math></span> where <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1354" display="inline"><mrow><mi>b</mi><mo>></mo><mn>0</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1355" display="inline"><mrow><mi>b</mi><mo>≠</mo><mn>1</mn></mrow></math></span>, are one-to-one; each value in the range corresponds to exactly one element in the domain. Therefore, <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1356" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span> implies <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1357" display="inline"><mrow><mi>x</mi><mo>=</mo><mi>y</mi></mrow><mo>.</mo></math></span> The converse is true because <em class="emphasis">f</em> is a function. This leads to the very important <span class="margin_term"><a class="glossterm">one-to-one property of exponential functions</a><span class="glossdef">Given <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1358" display="inline"><mrow><mi>b</mi><mo>></mo><mn>0</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1359" display="inline"><mrow><mi>b</mi><mo>≠</mo><mn>1</mn></mrow></math></span> we have <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1360" display="inline"><mrow><msup><mi>b</mi><mi>x</mi></msup><mo>=</mo><msup><mi>b</mi><mi>y</mi></msup></mrow></math></span> if and only if <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1361" display="inline"><mrow><mi>x</mi><mo>=</mo><mi>y</mi></mrow><mo>.</mo></math></span></span></span>:</p>
<p class="para block" id="fwk-redden-ch07_s05_s01_p02"><span class="informalequation">
<math xml:id="fwk-redden-ch07_m1362" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>b</mi><mi>x</mi></msup></mrow><mo>=</mo><mrow><msup><mi>b</mi><mi>y</mi></msup></mrow></mtd><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtd><mrow><mtext>if</mtext><mtext> </mtext><mtext>and</mtext><mtext> </mtext><mtext>only</mtext><mtext> </mtext><mtext>if</mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow></mtd><mtd columnalign="right"><mi>x</mi><mo>=</mo><mi>y</mi></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch07_s05_s01_p03">Use this property to solve special exponential equations where each side can be written in terms of the same base.</p>
<div class="callout block" id="fwk-redden-ch07_s05_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch07_s05_s01_p04">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1363" display="inline"><mrow><msup><mn>3</mn><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>27</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch07_s05_s01_p05">Begin by writing 27 as a power of 3.</p>
<p class="para" id="fwk-redden-ch07_s05_s01_p06"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1364" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mn>3</mn><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>27</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mn>3</mn><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mn>3</mn><mn>3</mn></msup></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s01_p07">Next apply the one-to-one property of exponential functions. In other words, set the exponents equal to each other and then simplify.</p>
<p class="para" id="fwk-redden-ch07_s05_s01_p08"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1365" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="right"><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr columnalign="right"><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr columnalign="right"><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s01_p09">Answer: 2</p>
</div>
<div class="callout block" id="fwk-redden-ch07_s05_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch07_s05_s01_p10">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1366" display="inline"><mrow><msup><mrow><mn>16</mn></mrow><mrow><mn>1</mn><mo>−</mo><mn>3</mn><mi>x</mi></mrow></msup><mo>=</mo><mn>2</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch07_s05_s01_p11">Begin by writing 16 as a power of 2 and then apply the power rule for exponents.</p>
<p class="para" id="fwk-redden-ch07_s05_s01_p12"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1367" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mn>16</mn></mrow><mrow><mn>1</mn><mo>−</mo><mn>3</mn><mi>x</mi></mrow></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mn>2</mn><mn>4</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>−</mo><mn>3</mn><mi>x</mi></mrow></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mn>2</mn><mrow><mn>4</mn><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mn>3</mn><mi>x</mi></mrow><mo>)</mo></mrow></mrow></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mn>2</mn><mn>1</mn></msup></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s01_p13">Now that the bases are the same we can set the exponents equal to each other and simplify.</p>
<p class="para" id="fwk-redden-ch07_s05_s01_p14"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1368" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>4</mn><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mn>3</mn><mi>x</mi></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>4</mn><mo>−</mo><mn>12</mn><mi>x</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mo>−</mo><mn>12</mn><mi>x</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>3</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo>−</mo><mn>3</mn></mrow><mrow><mo>−</mo><mn>12</mn></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s01_p15">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1369" display="inline"><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch07_s05_s01_n02a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch07_s05_s01_p16"><strong class="emphasis bold">Try this!</strong> Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1370" display="inline"><mrow><msup><mrow><mn>25</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></msup><mo>=</mo><mn>125</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s01_p17">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1371" display="inline"><mrow><mo>−</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></math></span></p>
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<p class="para editable block" id="fwk-redden-ch07_s05_s01_p19">In many cases we will not be able to equate the bases. For this reason we develop a second method for solving exponential equations. Consider the following equations:</p>
<p class="para block" id="fwk-redden-ch07_s05_s01_p20"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1372" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mn>3</mn><mn>2</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mn>3</mn><mstyle color="#007fbf"><mo>?</mo></mstyle></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>12</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mn>3</mn><mn>3</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>27</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch07_s05_s01_p21">We can see that the solution to <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1373" display="inline"><mrow><msup><mn>3</mn><mi>x</mi></msup><mo>=</mo><mn>12</mn></mrow></math></span> should be somewhere between 2 and 3. A graphical interpretation follows.</p>
<div class="informalfigure large block">
<img src="section_10/e840befd49bc2563c82fbeffe65c4a6d.png">
</div>
<p class="para block" id="fwk-redden-ch07_s05_s01_p23">To solve this we make use of fact that logarithms are one-to-one functions. Given <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1374" display="inline"><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>></mo><mn>0</mn></mrow></math></span> the <span class="margin_term"><a class="glossterm">one-to-one property of logarithms</a><span class="glossdef">Given <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1375" display="inline"><mrow><mi>b</mi><mo>></mo><mn>0</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1376" display="inline"><mrow><mi>b</mi><mo>≠</mo><mn>1</mn></mrow></math></span> where <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1377" display="inline"><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>></mo><mn>0</mn></mrow></math></span> we have <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1378" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mi>b</mi></msub><mtext> </mtext><mi>x</mi><mo>=</mo><msub><mrow><mi>log</mi></mrow><mi>b</mi></msub><mtext> </mtext><mi>y</mi></mrow></math></span> if and only if <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1379" display="inline"><mrow><mi>x</mi><mo>=</mo><mi>y</mi></mrow><mo>.</mo></math></span></span></span> follows:</p>
<p class="para block" id="fwk-redden-ch07_s05_s01_p24"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1380" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msub><mrow><mi>log</mi></mrow><mi>b</mi></msub><mtext> </mtext><mi>x</mi></mrow><mo>=</mo><mrow><msub><mrow><mi>log</mi></mrow><mi>b</mi></msub><mtext> </mtext><mi>y</mi></mrow></mtd><mtd><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext>if</mtext><mtext> </mtext><mtext>and</mtext><mtext> </mtext><mtext>only</mtext><mtext> </mtext><mtext>if</mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow></mtd><mtd columnalign="right"><mi>x</mi><mo>=</mo><mi>y</mi></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch07_s05_s01_p25">This property, as well as the properties of the logarithm, allows us to solve exponential equations. For example, to solve <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1381" display="inline"><mrow><msup><mn>3</mn><mi>x</mi></msup><mo>=</mo><mn>12</mn></mrow></math></span> apply the common logarithm to both sides and then use the properties of the logarithm to isolate the variable.</p>
<p class="para block" id="fwk-redden-ch07_s05_s01_p26"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1382" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mn>3</mn><mi>x</mi></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>12</mn></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>log</mi><mtext> </mtext><msup><mn>3</mn><mi>x</mi></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>log</mi><mtext> </mtext><mn>12</mn></mrow></mtd><mtd><mtext> </mtext><mtext> </mtext><mrow><mstyle color="#007fbf"><mi>O</mi><mi>n</mi><mi>e</mi><mtext>-</mtext><mi>t</mi><mi>o</mi><mtext>-</mtext><mi>o</mi><mi>n</mi><mi>e</mi><mtext> </mtext><mi>p</mi><mi>r</mi><mi>o</mi><mi>p</mi><mi>e</mi><mi>r</mi><mi>t</mi><mi>y</mi><mtext> </mtext><mi>o</mi><mi>f</mi><mtext> </mtext><mi>l</mi><mi>o</mi><mi>g</mi><mi>a</mi><mi>r</mi><mi>i</mi><mi>t</mi><mi>h</mi><mi>m</mi><mi>s</mi></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mi>log</mi><mtext> </mtext><mn>3</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>log</mi><mtext> </mtext><mn>12</mn></mrow></mtd><mtd><mtext> </mtext><mtext> </mtext><mrow><mstyle color="#007fbf"><mi>P</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi><mtext> </mtext><mi>r</mi><mi>u</mi><mi>l</mi><mi>e</mi><mtext> </mtext><mi>f</mi><mi>o</mi><mi>r</mi><mtext> </mtext><mi>l</mi><mi>o</mi><mi>g</mi><mi>a</mi><mi>r</mi><mi>i</mi><mi>t</mi><mi>h</mi><mi>m</mi><mi>s</mi></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mi>log</mi><mtext> </mtext><mn>12</mn></mrow><mrow><mi>log</mi><mtext> </mtext><mn>3</mn></mrow></mfrac></mrow></mtd><mtd><mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch07_s05_s01_p27">Approximating to four decimal places on a calculator.</p>
<p class="para block" id="fwk-redden-ch07_s05_s01_p28"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1383" display="block"><mrow><mi>x</mi><mo>=</mo><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mn>12</mn></mrow><mo>)</mo></mrow><mo>/</mo><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow><mo>≈</mo><mn>2.2619</mn></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch07_s05_s01_p29">An answer between 2 and 3 is what we expected. Certainly we can check by raising 3 to this power to verify that we obtain a good approximation of 12.</p>
<p class="para block" id="fwk-redden-ch07_s05_s01_p30"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1384" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>3</mn><mo>^</mo><mn>2.2618</mn><mo>≈</mo><mn>12</mn></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mtext>✓</mtext></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch07_s05_s01_p31">Note that we are <em class="emphasis bolditalic">not</em> multiplying both sides by “log”; we are applying the one-to-one property of logarithmic functions — which is often expressed as “<em class="emphasis">taking the log of both sides</em>.” The general steps for solving exponential equations are outlined in the following example.</p>
<div class="callout block" id="fwk-redden-ch07_s05_s01_n03">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch07_s05_s01_p32">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1385" display="inline"><mrow><msup><mn>5</mn><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mn>2</mn><mo>=</mo><mn>9</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<ul class="itemizedlist" id="fwk-redden-ch07_s05_s01_l01" mark="none">
<li>
<p class="para"><strong class="emphasis bold">Step 1:</strong> Isolate the exponential expression.</p>
<p class="para"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1386" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mn>5</mn><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mn>2</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mn>5</mn><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn></mtd></mtr></mtable></mrow></math></span></p>
</li>
<li>
<p class="para"><strong class="emphasis bold">Step 2:</strong> Take the logarithm of both sides. In this case, we will take the common logarithm of both sides so that we can approximate our result on a calculator.</p>
<p class="para"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1387" display="block"><mrow><mi>log</mi><mtext> </mtext><msup><mn>5</mn><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mi>log</mi><mtext> </mtext><mn>7</mn></mrow></math></span></p>
</li>
<li>
<p class="para"><strong class="emphasis bold">Step 3:</strong> Apply the power rule for logarithms and then solve.</p>
<p class="para"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1388" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>log</mi><mtext> </mtext><msup><mn>5</mn><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>log</mi><mtext> </mtext><mn>7</mn></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>log</mi><mtext> </mtext><mn>5</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>log</mi><mtext> </mtext><mn>7</mn></mrow></mtd><mtd><mtext> </mtext><mtext> </mtext><mrow><mstyle color="#007fbf"><mi>D</mi><mi>i</mi><mi>s</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mo>.</mo></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mi>log</mi><mtext> </mtext><mn>5</mn><mo>−</mo><mi>log</mi><mtext> </mtext><mn>5</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>log</mi><mtext> </mtext><mn>7</mn></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mi>log</mi><mtext> </mtext><mn>5</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>log</mi><mtext> </mtext><mn>5</mn><mo>+</mo><mi>log</mi><mtext> </mtext><mn>7</mn></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mi>log</mi><mtext> </mtext><mn>5</mn><mo>+</mo><mi>log</mi><mtext> </mtext><mn>7</mn></mrow><mrow><mn>2</mn><mi>log</mi><mtext> </mtext><mn>5</mn></mrow></mfrac></mrow></mtd><mtd><mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
</li>
</ul>
<p class="para" id="fwk-redden-ch07_s05_s01_p33">This is an irrational number which can be approximated using a calculator. Take care to group the numerator and the product in the denominator when entering this into your calculator. To do this, make use of the parenthesis buttons <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1389" display="inline"><mrow><mtable frame="solid" columnspacing="0.1em"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mtext> </mtext></mrow></mtable></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1390" display="inline"><mrow><mtable frame="solid" columnspacing="0.1em"><mrow><mtext> </mtext><mo stretchy="false">)</mo><mtext> </mtext></mrow></mtable></mrow></math></span> :</p>
<p class="para" id="fwk-redden-ch07_s05_s01_p34"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1391" display="block"><mrow><mi>x</mi><mo>=</mo><mrow><mo>(</mo><mrow><mi>log</mi><mtext> </mtext><mn>5</mn><mo>+</mo><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mn>7</mn><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>/</mo><mrow><mo>(</mo><mrow><mn>2</mn><mo>*</mo><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>≈</mo><mn>1.1045</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s01_p35">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1392" display="inline"><mrow><mfrac><mrow><mi>log</mi><mtext> </mtext><mn>5</mn><mo>+</mo><mi>log</mi><mtext> </mtext><mn>7</mn></mrow><mrow><mn>2</mn><mi>log</mi><mtext> </mtext><mn>5</mn></mrow></mfrac><mo>≈</mo><mn>1.1045</mn></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch07_s05_s01_n04">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch07_s05_s01_p36">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1393" display="inline"><mrow><msup><mi>e</mi><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></msup><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch07_s05_s01_p37">The exponential function is already isolated and the base is <em class="emphasis">e</em>. Therefore, we choose to apply the natural logarithm to both sides.</p>
<p class="para" id="fwk-redden-ch07_s05_s01_p38"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1394" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>e</mi><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>ln</mi><mtext> </mtext><msup><mi>e</mi><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>ln</mi><mtext> </mtext><mn>1</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s01_p39">Apply the power rule for logarithms and then simplify.</p>
<p class="para" id="fwk-redden-ch07_s05_s01_p40"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1395" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>ln</mi><mtext> </mtext><msup><mi>e</mi><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>ln</mi><mtext> </mtext><mn>1</mn></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow><mi>ln</mi><mtext> </mtext><mi>e</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>ln</mi><mtext> </mtext><mn>1</mn></mrow></mtd><mtd><mtext> </mtext><mtext> </mtext><mrow><mstyle color="#007fbf"><mi>R</mi><mi>e</mi><mi>c</mi><mi>a</mi><mi>l</mi><mi>l</mi><mtext> </mtext><mtext>ln</mtext><mtext> </mtext><mi>e</mi><mo>=</mo><mn>1</mn><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mtext>ln</mtext><mn>1</mn><mo>=</mo><mn>0</mn><mo>.</mo></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>⋅</mo><mn>1</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>3</mn><mn>5</mn></mfrac></mrow></mtd><mtd><mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s01_p41">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1396" display="inline"><mrow><mo>−</mo><mfrac><mn>3</mn><mn>5</mn></mfrac></mrow></math></span></p>
</div>
<p class="para block" id="fwk-redden-ch07_s05_s01_p42">On most calculators there are only two logarithm buttons, the common logarithm <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1397" display="inline"><mrow><mtable frame="solid" columnspacing="0.1em"><mrow><mi>L</mi><mi>O</mi><mi>G</mi></mrow></mtable></mrow></math></span> and the natural logarithm <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1398" display="inline"><mrow><mtable frame="solid" columnspacing="0.1em"><mrow><mi>L</mi><mi>N</mi></mrow></mtable></mrow><mo>.</mo></math></span> If we want to approximate <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1399" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>3</mn></msub><mtext> </mtext><mn>10</mn></mrow></math></span> we have to somehow change this base to 10 or <em class="emphasis">e</em>. The idea begins by rewriting the logarithmic function <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1400" display="inline"><mrow><mi>y</mi><mo>=</mo><msub><mrow><mi>log</mi></mrow><mi>a</mi></msub><mtext> </mtext><mi>x</mi></mrow></math></span>, in exponential form.</p>
<p class="para block" id="fwk-redden-ch07_s05_s01_p43"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1401" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msub><mrow><mi>log</mi></mrow><mi>a</mi></msub><mtext> </mtext><mi>x</mi></mrow><mo>=</mo><mi>y</mi></mtd><mtd><mstyle color="#007fbf"><mtext> </mtext><mtext> </mtext><mo>⇒</mo><mtext> </mtext><mtext> </mtext></mstyle></mtd><mtd columnalign="right"><mi>x</mi><mo>=</mo><mrow><msup><mi>a</mi><mi>y</mi></msup></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch07_s05_s01_p44">Here <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1402" display="inline"><mrow><mi>x</mi><mo>></mo><mn>0</mn></mrow></math></span> and so we can apply the one-to-one property of logarithms. Apply the logarithm base <em class="emphasis">b</em> to both sides of the function in exponential form.</p>
<p class="para block" id="fwk-redden-ch07_s05_s01_p45"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1403" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>a</mi><mi>y</mi></msup></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msub><mrow><mi>log</mi></mrow><mi>b</mi></msub><mtext> </mtext><mi>x</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mrow><mi>log</mi></mrow><mi>b</mi></msub><mtext> </mtext><msup><mi>a</mi><mi>y</mi></msup></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch07_s05_s01_p46">And then solve for <em class="emphasis">y</em>.</p>
<p class="para block" id="fwk-redden-ch07_s05_s01_p47"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1404" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msub><mrow><mi>log</mi></mrow><mi>b</mi></msub><mtext> </mtext><mi>x</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>y</mi><msub><mrow><mi>log</mi></mrow><mi>b</mi></msub><mtext> </mtext><mi>a</mi></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><msub><mrow><mi>log</mi></mrow><mi>b</mi></msub><mtext> </mtext><mi>x</mi></mrow><mrow><msub><mrow><mi>log</mi></mrow><mi>b</mi></msub><mtext> </mtext><mi>a</mi></mrow></mfrac></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>y</mi></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch07_s05_s01_p48">Replace <em class="emphasis">y</em> into the original function and we have the very important <span class="margin_term"><a class="glossterm">change of base formula</a><span class="glossdef"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1405" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mi>a</mi></msub><mtext> </mtext><mi>x</mi><mo>=</mo><mfrac><mrow><msub><mrow><mi>log</mi></mrow><mi>b</mi></msub><mtext> </mtext><mi>x</mi></mrow><mrow><msub><mrow><mi>log</mi></mrow><mi>b</mi></msub><mtext> </mtext><mi>a</mi></mrow></mfrac></mrow></math></span>; we can write any base-<em class="emphasis">a</em> logarithm in terms of base-<em class="emphasis">b</em> logarithms using this formula.</span></span>:</p>
<p class="para block" id="fwk-redden-ch07_s05_s01_p49"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1406" display="block"><mrow><msub><mrow><mi>log</mi></mrow><mi>a</mi></msub><mtext> </mtext><mi>x</mi><mo>=</mo><mfrac><mrow><msub><mrow><mi>log</mi></mrow><mi>b</mi></msub><mtext> </mtext><mi>x</mi></mrow><mrow><msub><mrow><mi>log</mi></mrow><mi>b</mi></msub><mtext> </mtext><mi>a</mi></mrow></mfrac></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch07_s05_s01_p50">We can use this to approximate <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1407" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>3</mn></msub><mtext> </mtext><mn>10</mn></mrow></math></span> as follows.</p>
<p class="para block" id="fwk-redden-ch07_s05_s01_p51"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1408" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msub><mrow><mi>log</mi></mrow><mn>3</mn></msub><mtext> </mtext><mn>10</mn></mrow><mo>=</mo><mrow><mfrac><mrow><mi>log</mi><mtext> </mtext><mn>10</mn></mrow><mrow><mi>log</mi><mtext> </mtext><mn>3</mn></mrow></mfrac><mo>≈</mo><mn>2.0959</mn></mrow></mtd><mtd><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mrow><mi>o</mi><mi>r</mi></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mtd><mtd columnalign="right"><mrow><msub><mrow><mi>log</mi></mrow><mn>3</mn></msub><mtext> </mtext><mn>10</mn></mrow><mo>=</mo><mrow><mfrac><mrow><mi>ln</mi><mtext> </mtext><mn>10</mn></mrow><mrow><mi>ln</mi><mtext> </mtext><mn>3</mn></mrow></mfrac><mo>≈</mo><mn>2.0959</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch07_s05_s01_p52">Notice that the result is independent of the choice of base. In words, we can approximate the logarithm of any given base on a calculator by dividing the logarithm of the argument by the logarithm of that given base.</p>
<div class="callout block" id="fwk-redden-ch07_s05_s01_n05">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch07_s05_s01_p53">Approximate <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1409" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>7</mn></msub><mtext> </mtext><mn>120</mn></mrow></math></span> the nearest hundredth.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch07_s05_s01_p54">Apply the change of base formula and use a calculator.</p>
<p class="para" id="fwk-redden-ch07_s05_s01_p55"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1410" display="block"><mrow><msub><mrow><mi>log</mi></mrow><mn>7</mn></msub><mtext> </mtext><mn>120</mn><mo>=</mo><mfrac><mrow><mi>log</mi><mtext> </mtext><mn>120</mn></mrow><mrow><mi>log</mi><mtext> </mtext><mn>7</mn></mrow></mfrac></mrow></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s01_p56">On a calculator,</p>
<p class="para" id="fwk-redden-ch07_s05_s01_p57"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1411" display="block"><mrow><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mn>120</mn></mrow><mo>)</mo></mrow><mo>/</mo><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mn>7</mn><mo>)</mo></mrow><mo>≈</mo><mn>2.46</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s01_p58">Answer: 2.46</p>
</div>
<div class="callout block" id="fwk-redden-ch07_s05_s01_n05a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch07_s05_s01_p59"><strong class="emphasis bold">Try this!</strong> Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1412" display="inline"><mrow><msup><mn>2</mn><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>4</mn><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span> Give the exact and approximate answer rounded to four decimal places.</p>
<p class="para" id="fwk-redden-ch07_s05_s01_p60">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1413" display="inline"><mrow><mfrac><mrow><mi>log</mi><mtext> </mtext><mn>5</mn><mo>−</mo><mi>log</mi><mtext> </mtext><mn>2</mn></mrow><mrow><mn>3</mn><mi>log</mi><mtext> </mtext><mn>2</mn></mrow></mfrac><mo>≈</mo><mn>0.4406</mn></mrow></math></span></p>
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<div class="section" id="fwk-redden-ch07_s05_s02" version="5.0" lang="en">
<h2 class="title editable block">Solving Logarithmic Equations</h2>
<p class="para editable block" id="fwk-redden-ch07_s05_s02_p01">A <span class="margin_term"><a class="glossterm">logarithmic equation</a><span class="glossdef">An equation that involves a logarithm with a variable argument.</span></span> is an equation that involves a logarithm with a variable argument. Some logarithmic equations can be solved using the one-to-one property of logarithms. This is true when a single logarithm with the same base can be obtained on both sides of the equal sign.</p>
<div class="callout block" id="fwk-redden-ch07_s05_s02_n01">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch07_s05_s02_p02">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1414" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>−</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch07_s05_s02_p03">We can obtain two equal logarithms base 2 by adding <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1415" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span> to both sides of the equation.</p>
<p class="para" id="fwk-redden-ch07_s05_s02_p04"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1416" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>−</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s02_p05">Here the bases are the same and so we can apply the one-to-one property and set the arguments equal to each other.</p>
<p class="para" id="fwk-redden-ch07_s05_s02_p06"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1417" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s02_p07">Checking <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1418" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>3</mn></mrow></math></span> in the original equation:</p>
<p class="para" id="fwk-redden-ch07_s05_s02_p08"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1419" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>2</mn><mrow><mo>(</mo><mstyle color="#007f3f"><mn>3</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mrow><mo>(</mo><mstyle color="#007f3f"><mn>3</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mn>1</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mn>1</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>0</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mtext>✓</mtext></mstyle></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s02_p09">Answer: 3</p>
</div>
<p class="para block" id="fwk-redden-ch07_s05_s02_p10">When solving logarithmic equations the check is very important because extraneous solutions can be obtained. The properties of the logarithm only apply for values in the domain of the given logarithm. And when working with variable arguments, such as <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1420" display="inline"><mrow><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span>, the value of <em class="emphasis">x</em> is not known until the end of this process. The logarithmic expression <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1421" display="inline"><mrow><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span> is only defined for values <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1422" display="inline"><mrow><mi>x</mi><mo>></mo><mn>2</mn></mrow><mo>.</mo></math></span></p>
<div class="callout block" id="fwk-redden-ch07_s05_s02_n02">
<h3 class="title">Example 7</h3>
<p class="para" id="fwk-redden-ch07_s05_s02_p11">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1423" display="inline"><mrow><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow><mo>=</mo><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch07_s05_s02_p12">Apply the one-to-one property of logarithms (set the arguments equal to each other) and then solve for <em class="emphasis">x</em>.</p>
<p class="para" id="fwk-redden-ch07_s05_s02_p13"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1424" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s02_p14">When performing the check we encounter a logarithm of a negative number:</p>
<p class="para" id="fwk-redden-ch07_s05_s02_p15"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1425" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>1</mn></mstyle><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mtext> </mtext><mtext> </mtext><mrow><mstyle color="#007fbf"><mi>U</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>f</mi><mi>i</mi><mi>n</mi><mi>e</mi><mi>d</mi></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s02_p16">Try this on a calculator, what does it say? Here <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1426" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></math></span> is not in the domain of <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1427" display="inline"><mrow><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span> Therefore our only possible solution is extraneous and we conclude that there are no solutions to this equation.</p>
<p class="para" id="fwk-redden-ch07_s05_s02_p17">Answer: No solution, Ø.</p>
</div>
<p class="para editable block" id="fwk-redden-ch07_s05_s02_p18"><strong class="emphasis bold">Caution:</strong> Solving logarithmic equations sometimes leads to extraneous solutions — we must check our answers.</p>
<div class="callout block" id="fwk-redden-ch07_s05_s02_n02a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch07_s05_s02_p19"><strong class="emphasis bold">Try this!</strong> Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1428" display="inline"><mrow><mi>ln</mi><mtext> </mtext><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>15</mn></mrow><mo>)</mo></mrow><mo>−</mo><mi>ln</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s02_p20">Answer: 5</p>
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<p class="para block" id="fwk-redden-ch07_s05_s02_p22">In many cases we will not be able to obtain two equal logarithms. To solve such equations we make use of the definition of the logarithm. If <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1429" display="inline"><mrow><mi>b</mi><mo>></mo><mn>0</mn></mrow></math></span>, where <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1430" display="inline"><mrow><mi>b</mi><mo>≠</mo><mn>1</mn></mrow></math></span>, then <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1431" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mi>b</mi></msub><mtext> </mtext><mi>x</mi><mo>=</mo><mi>y</mi></mrow></math></span> implies that <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1432" display="inline"><mrow><msup><mi>b</mi><mi>y</mi></msup><mo>=</mo><mi>x</mi></mrow><mo>.</mo></math></span> Consider the following common logarithmic equations (base 10),</p>
<p class="para block" id="fwk-redden-ch07_s05_s02_p23"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1433" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>log</mi><mtext> </mtext><mi>x</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd><mtext> </mtext><mtext> </mtext><mo>⇒</mo><mtext> </mtext><mtext> </mtext></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd><mtd columnalign="right"><mtext> </mtext><mtext> </mtext><mrow><mstyle color="#007fbf"><mi>B</mi><mi>e</mi><mi>c</mi><mi>a</mi><mi>u</mi><mi>s</mi><mi>e</mi><mtext> </mtext><mtext> </mtext><msup><mrow><mn>10</mn></mrow><mn>0</mn></msup></mstyle></mrow></mtd><mtd><mstyle color="#007fbf"><mo>=</mo></mstyle></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mn>1</mn><mo>.</mo></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>log</mi><mtext> </mtext><mstyle color="#007fbf"><mi>x</mi></mstyle></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>0.5</mn></mrow></mtd><mtd><mtext> </mtext><mtext> </mtext><mo>⇒</mo><mtext> </mtext><mtext> </mtext></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mo>?</mo></mstyle></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>log</mi><mtext> </mtext><mi>x</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd><mtd><mtext> </mtext><mtext> </mtext><mo>⇒</mo><mtext> </mtext><mtext> </mtext></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>10</mn></mrow></mtd><mtd columnalign="right"><mtext> </mtext><mtext> </mtext><mrow><mstyle color="#007fbf"><mi>B</mi><mi>e</mi><mi>c</mi><mi>a</mi><mi>u</mi><mi>s</mi><mi>e</mi><mtext> </mtext><mtext> </mtext><msup><mrow><mn>10</mn></mrow><mn>1</mn></msup></mstyle></mrow></mtd><mtd><mstyle color="#007fbf"><mo>=</mo></mstyle></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mn>10</mn><mo>.</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch07_s05_s02_p24">We can see that the solution to <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1434" display="inline"><mrow><mi>log</mi><mtext> </mtext><mi>x</mi><mo>=</mo><mn>0.5</mn></mrow></math></span> will be somewhere between 1 and 10. A graphical interpretation follows.</p>
<div class="informalfigure large block">
<img src="section_10/9d92e28c6a92ce9f4f32bcb4e6e228d4.png">
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<p class="para editable block" id="fwk-redden-ch07_s05_s02_p26">To find <em class="emphasis">x</em> we can apply the definition as follows.</p>
<p class="para block" id="fwk-redden-ch07_s05_s02_p27"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1435" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msub><mrow><mi>log</mi></mrow><mrow><mn>10</mn></mrow></msub><mtext> </mtext><mi>x</mi></mrow><mo>=</mo><mrow><mn>0.5</mn></mrow></mtd><mtd><mtext> </mtext><mtext> </mtext><mo>⇒</mo><mtext> </mtext><mtext> </mtext></mtd><mtd columnalign="right"><mrow><msup><mrow><mn>10</mn></mrow><mrow><mn>0.5</mn></mrow></msup></mrow><mo>=</mo><mi>x</mi></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch07_s05_s02_p28">This can be approximated using a calculator,</p>
<p class="para block" id="fwk-redden-ch07_s05_s02_p29"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1436" display="block"><mrow><mi>x</mi><mo>=</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>0.5</mn></mrow></msup><mo>=</mo><mn>10</mn><mo>^</mo><mn>0.5</mn><mo>≈</mo><mn>3.1623</mn></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch07_s05_s02_p30">An answer between 1 and 10 is what we expected. Check this on a calculator.</p>
<p class="para block" id="fwk-redden-ch07_s05_s02_p31"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1437" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>log</mi><mtext> </mtext><mn>3.1623</mn><mo>≈</mo><mn>5</mn></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<div class="callout block" id="fwk-redden-ch07_s05_s02_n03">
<h3 class="title">Example 8</h3>
<p class="para" id="fwk-redden-ch07_s05_s02_p32">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1438" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>3</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>2</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch07_s05_s02_p33">Apply the definition of the logarithm.</p>
<p class="para" id="fwk-redden-ch07_s05_s02_p34"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1439" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msub><mrow><mi>log</mi></mrow><mn>3</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mn>2</mn></mtd><mtd><mstyle color="#007fbf"><mtext> </mtext><mtext> </mtext><mo>⇒</mo><mtext> </mtext><mtext> </mtext></mstyle></mtd><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>=</mo><mrow><msup><mn>3</mn><mn>2</mn></msup></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s02_p35">Solve the resulting equation.</p>
<p class="para" id="fwk-redden-ch07_s05_s02_p36"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1440" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>14</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s02_p37">Check.</p>
<p class="para" id="fwk-redden-ch07_s05_s02_p38"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1441" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msub><mi>log</mi><mn>3</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>2</mn><mrow><mo>(</mo><mstyle color="#007f3f"><mn>7</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mtd><mtd><mover><mo>=</mo><mo>?</mo></mover></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><msub><mi>log</mi><mn>3</mn></msub><mtext> </mtext><mrow><mo>(</mo><mn>9</mn><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mtext>✓</mtext></mstyle></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s02_p39">Answer: 7</p>
</div>
<p class="para editable block" id="fwk-redden-ch07_s05_s02_p40">In order to apply the definition, we will need to rewrite logarithmic expressions as a single logarithm with coefficient 1.The general steps for solving logarithmic equations are outlined in the following example.</p>
<div class="callout block" id="fwk-redden-ch07_s05_s02_n04">
<h3 class="title">Example 9</h3>
<p class="para" id="fwk-redden-ch07_s05_s02_p41">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1442" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<ul class="itemizedlist" id="fwk-redden-ch07_s05_s02_l01" mark="none">
<li>
<p class="para"><strong class="emphasis bold">Step 1:</strong> Write all logarithmic expressions as a single logarithm with coefficient 1. In this case, apply the product rule for logarithms.</p>
<p class="para"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1443" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>[</mo><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr></mtable></mrow></math></span></p>
</li>
<li>
<p class="para"><strong class="emphasis bold">Step 2:</strong> Use the definition and rewrite the logarithm in exponential form.</p>
<p class="para"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1444" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>[</mo><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mrow><mo>=</mo><mn>1</mn></mtd><mtd><mstyle color="#007fbf"><mtext> </mtext><mtext> </mtext><mo>⇒</mo><mtext> </mtext><mtext> </mtext></mstyle></mtd><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mn>2</mn><mn>1</mn></msup></mrow></mtd></mtr></mtable></mrow></math></span></p>
</li>
<li>
<p class="para"><strong class="emphasis bold">Step 3:</strong> Solve the resulting equation. Here we can solve by factoring.</p>
<p class="para"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1445" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>6</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mrow><mtext>or</mtext></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mtd><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd><mtd><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr></mtable></mrow></math></span></p>
</li>
<li>
<p class="para"><strong class="emphasis bold">Step 4:</strong> Check. This step is required.</p>
<p class="para"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><em class="emphasis">Check</em> <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1446" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>4</mn></mrow></math></span></p></th>
<th align="center"><p class="para"><em class="emphasis">Check</em> <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1447" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></math></span></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1448" display="inline"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mn>4</mn></mstyle><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mn>4</mn></mstyle><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>1</mn><mo>+</mo><mn>0</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mtext>✓</mtext></mstyle></mtd></mtr></mtable></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1449" display="inline"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mn>1</mn></mstyle><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mn>1</mn></mstyle><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#ff0000"><mo>✗</mo></mstyle></mtd></mtr></mtable></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
</li>
</ul>
<p class="para" id="fwk-redden-ch07_s05_s02_p42">In this example, <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1450" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></math></span> is not in the domain of the given logarithmic expression and is extraneous. The only solution is <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1451" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>4</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s02_p43">Answer: 4</p>
</div>
<div class="callout block" id="fwk-redden-ch07_s05_s02_n05">
<h3 class="title">Example 10</h3>
<p class="para" id="fwk-redden-ch07_s05_s02_p44">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1452" display="inline"><mrow><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>15</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>1</mn><mo>=</mo><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch07_s05_s02_p45">Begin by writing all logarithmic expressions on one side and constants on the other.</p>
<p class="para" id="fwk-redden-ch07_s05_s02_p46"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1453" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>15</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>15</mn></mrow><mo>)</mo></mrow><mo>−</mo><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s02_p47">Apply the quotient rule for logarithms as a means to obtain a single logarithm with coefficient 1.</p>
<p class="para" id="fwk-redden-ch07_s05_s02_p48"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1454" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>15</mn></mrow><mo>)</mo></mrow><mo>−</mo><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mfrac><mrow><mi>x</mi><mo>+</mo><mn>15</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s02_p49">This is a common logarithm; therefore use 10 as the base when applying the definition.</p>
<p class="para" id="fwk-redden-ch07_s05_s02_p50"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1455" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mi>x</mi><mo>+</mo><mn>15</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow></mfrac></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mn>10</mn></mrow><mn>1</mn></msup></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mo>+</mo><mn>15</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>10</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mo>+</mo><mn>15</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>10</mn><mi>x</mi><mo>+</mo><mn>60</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mo>−</mo><mn>9</mn><mi>x</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>45</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>5</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s02_p51">Check.</p>
<p class="para" id="fwk-redden-ch07_s05_s02_p52"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1456" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>15</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mo>−</mo><mstyle color="#007f3f"><mn>5</mn></mstyle><mo>+</mo><mn>15</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mo>−</mo><mstyle color="#007f3f"><mn>5</mn></mstyle><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>log</mi><mtext> </mtext><mn>10</mn><mo>−</mo><mn>1</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>log</mi><mtext> </mtext><mn>1</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>1</mn><mo>−</mo><mn>1</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>0</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mtext>✓</mtext></mstyle></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s02_p53">Answer: −5</p>
</div>
<div class="callout block" id="fwk-redden-ch07_s05_s02_n05a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch07_s05_s02_p54"><strong class="emphasis bold">Try this!</strong> Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1457" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s02_p55">Answer: 2</p>
<div class="mediaobject">
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</div>
</div>
<div class="callout block" id="fwk-redden-ch07_s05_s02_n06">
<h3 class="title">Example 11</h3>
<p class="para" id="fwk-redden-ch07_s05_s02_p57">Find the inverse: <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1458" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch07_s05_s02_p58">Begin by replacing the function notation <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1459" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> with <em class="emphasis">y</em>.</p>
<p class="para" id="fwk-redden-ch07_s05_s02_p59"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1460" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s02_p60">Interchange <em class="emphasis">x</em> and <em class="emphasis">y</em> and then solve for <em class="emphasis">y</em>.</p>
<p class="para" id="fwk-redden-ch07_s05_s02_p61"><span class="informalequation"><math xml:id="fwk-redden-ch07_m1461" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>3</mn><mi>y</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mstyle color="#007fbf"><mtext> </mtext><mtext> </mtext><mo>⇒</mo><mtext> </mtext><mtext> </mtext></mstyle></mtd><mtd columnalign="right"><mrow><mn>3</mn><mi>y</mi><mo>−</mo><mn>4</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mn>2</mn><mi>x</mi></msup></mrow></mtd></mtr><mtr columnalign="left"><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd columnalign="right"><mrow><mn>3</mn><mi>y</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mn>2</mn><mi>x</mi></msup><mo>+</mo><mn>4</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><msup><mn>2</mn><mi>x</mi></msup><mo>+</mo><mn>4</mn></mrow><mn>3</mn></mfrac></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch07_s05_s02_p62">The resulting function is the inverse of <em class="emphasis">f</em>. Present the answer using function notation.</p>
<p class="para" id="fwk-redden-ch07_s05_s02_p63">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1462" display="inline"><mrow><msup><mi>f</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><msup><mn>2</mn><mi>x</mi></msup><mo>+</mo><mn>4</mn></mrow><mn>3</mn></mfrac></mrow></math></span></p>
</div>
<div class="key_takeaways editable block" id="fwk-redden-ch07_s05_s02_n07">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch07_s05_s02_l02" mark="bullet">
<li>If each side of an exponential equation can be expressed using the same base, then equate the exponents and solve.</li>
<li>To solve a general exponential equation, first isolate the exponential expression and then apply the appropriate logarithm to both sides. This allows us to use the properties of logarithms to solve for the variable.</li>
<li>The change of base formula allows us to use a calculator to calculate logarithms. The logarithm of a number is equal to the common logarithm of the number divided by the common logarithm of the given base.</li>
<li>If a single logarithm with the same base can be isolated on each side of an equation, then equate the arguments and solve.</li>
<li>To solve a general logarithmic equation, first isolate the logarithm with coefficient 1 and then apply the definition. Solve the resulting equation.</li>
<li>The steps for solving logarithmic equations sometimes produce extraneous solutions. Therefore, the check is required.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch07_s05_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch07_s05_qs01_qd01">
<h3 class="title">Part A: Solving Exponential Equations</h3>
<ol class="qandadiv" id="fwk-redden-ch07_s05_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch07_s05_qs01_p01"><strong class="emphasis bold">Solve using the one-to-one property of exponential functions.</strong></p>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa01">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p02"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1463" display="inline"><mrow><msup><mn>3</mn><mi>x</mi></msup><mo>=</mo><mn>81</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa02">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p04"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1464" display="inline"><mrow><msup><mn>2</mn><mrow><mo>−</mo><mi>x</mi></mrow></msup><mo>=</mo><mn>16</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa03">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p06"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1465" display="inline"><mrow><msup><mn>5</mn><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>25</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa04">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p08"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1466" display="inline"><mrow><msup><mn>3</mn><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></msup><mo>=</mo><mn>27</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa05">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p10"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1467" display="inline"><mrow><msup><mn>2</mn><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>=</mo><mn>16</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa06">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p12"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1469" display="inline"><mrow><msup><mn>2</mn><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></msup><mo>=</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa07">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p14"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1471" display="inline"><mrow><msup><mrow><mn>81</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa08">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p16"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1473" display="inline"><mrow><msup><mrow><mn>64</mn></mrow><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa09">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p18"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1475" display="inline"><mrow><msup><mn>9</mn><mrow><mn>2</mn><mo>−</mo><mn>3</mn><mi>x</mi></mrow></msup><mo>−</mo><mn>27</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa10">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p20"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1477" display="inline"><mrow><msup><mn>8</mn><mrow><mn>1</mn><mo>−</mo><mn>5</mn><mi>x</mi></mrow></msup><mo>−</mo><mn>32</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa11">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p22"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1479" display="inline"><mrow><msup><mrow><mn>16</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></msup><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa12">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p24"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1481" display="inline"><mrow><msup><mn>4</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>64</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa13">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p26"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1482" display="inline"><mrow><msup><mn>9</mn><mrow><mi>x</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></msup><mo>=</mo><mn>81</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa14">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p28"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1483" display="inline"><mrow><msup><mn>4</mn><mrow><mi>x</mi><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></msup><mo>=</mo><mn>64</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa15">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p30"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1485" display="inline"><mrow><msup><mrow><mn>100</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></msup><mo>−</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>7</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></msup><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa16">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p32"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1487" display="inline"><mrow><msup><mi>e</mi><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></msup><mo>−</mo><mi>e</mi><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch07_s05_qs01_qd01_qd02" start="17">
<p class="para" id="fwk-redden-ch07_s05_qs01_p34"><strong class="emphasis bold">Solve. Give the exact answer and the approximate answer rounded to the nearest thousandth.</strong></p>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa17">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p35"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1489" display="inline"><mrow><msup><mn>3</mn><mi>x</mi></msup><mo>=</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa18">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p37"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1491" display="inline"><mrow><msup><mn>7</mn><mi>x</mi></msup><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa19">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p39"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1493" display="inline"><mrow><msup><mn>4</mn><mi>x</mi></msup><mo>=</mo><mn>9</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa20">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p41"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1495" display="inline"><mrow><msup><mn>2</mn><mi>x</mi></msup><mo>=</mo><mn>10</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa21">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p43"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1497" display="inline"><mrow><msup><mn>5</mn><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></msup><mo>=</mo><mn>13</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa22">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p45"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1499" display="inline"><mrow><msup><mn>3</mn><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow></msup><mo>=</mo><mn>17</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa23">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p47"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1501" display="inline"><mrow><msup><mn>7</mn><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></msup><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa24">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p49"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1503" display="inline"><mrow><msup><mn>3</mn><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>9</mn></mrow></msup><mo>=</mo><mn>11</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa25">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p51"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1505" display="inline"><mrow><msup><mn>5</mn><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></msup><mo>+</mo><mn>6</mn><mo>=</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa26">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p53"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1506" display="inline"><mrow><msup><mrow><mn>10</mn></mrow><mrow><mn>7</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>2</mn><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa27">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p55"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1508" display="inline"><mrow><msup><mi>e</mi><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>5</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa28">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p57"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1510" display="inline"><mrow><msup><mi>e</mi><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>10</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa29">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p59"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1512" display="inline"><mrow><msup><mn>6</mn><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>3</mn><mo>=</mo><mn>7</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa30">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p61"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1514" display="inline"><mrow><mn>8</mn><mo>−</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>9</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow></msup><mo>=</mo><mn>9</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa31">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p63"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1515" display="inline"><mrow><mn>15</mn><mo>−</mo><msup><mi>e</mi><mrow><mn>3</mn><mi>x</mi></mrow></msup><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa32">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p65"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1517" display="inline"><mrow><mn>7</mn><mo>+</mo><msup><mi>e</mi><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>10</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa33">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p67"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1519" display="inline"><mrow><mn>7</mn><mo>−</mo><mn>9</mn><msup><mi>e</mi><mrow><mo>−</mo><mi>x</mi></mrow></msup><mo>=</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa34">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p69"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1521" display="inline"><mrow><mn>3</mn><mo>−</mo><mn>6</mn><msup><mi>e</mi><mrow><mo>−</mo><mi>x</mi></mrow></msup><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa35">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p71"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1523" display="inline"><mrow><msup><mn>5</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></msup><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa36">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p73"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1525" display="inline"><mrow><msup><mn>3</mn><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi></mrow></msup><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa37">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p75"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1527" display="inline"><mrow><mn>100</mn><msup><mi>e</mi><mrow><mn>27</mn><mi>x</mi></mrow></msup><mo>=</mo><mn>50</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa38">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p77"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1529" display="inline"><mrow><mn>6</mn><msup><mi>e</mi><mrow><mn>12</mn><mi>x</mi></mrow></msup><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa39">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch07_m1531" display="block"><mrow><mfrac><mn>3</mn><mrow><mn>1</mn><mo>+</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>x</mi></mrow></msup></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa40">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch07_m1533" display="block"><mrow><mfrac><mn>2</mn><mrow><mn>1</mn><mo>+</mo><mn>3</mn><msup><mi>e</mi><mrow><mo>−</mo><mi>x</mi></mrow></msup></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch07_s05_qs01_qd01_qd03" start="41">
<p class="para" id="fwk-redden-ch07_s05_qs01_p83"><strong class="emphasis bold">Find the <em class="emphasis">x</em>- and <em class="emphasis">y</em>-intercepts of the given function.</strong></p>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa41">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p84"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1535" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mn>3</mn><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa42">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p86"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1538" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mn>2</mn><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa43">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p88"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1541" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mn>10</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa44">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p90"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1543" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>4</mn><mi>x</mi></mrow></msup><mo>−</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa45">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p92"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1546" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>e</mi><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa46">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p94"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1548" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>e</mi><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></msup><mo>−</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch07_s05_qs01_qd01_qd04" start="47">
<p class="para" id="fwk-redden-ch07_s05_qs01_p96"><strong class="emphasis bold">Use a <em class="emphasis">u</em>-substitution to solve the following.</strong></p>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa47">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p97"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1551" display="inline"><mrow><msup><mn>3</mn><mrow><mn>2</mn><mi>x</mi></mrow></msup><mo>−</mo><msup><mn>3</mn><mi>x</mi></msup><mo>−</mo><mn>6</mn><mo>=</mo><mn>0</mn></mrow></math></span> (Hint: Let <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1552" display="inline"><mrow><mi>u</mi><mo>=</mo><msup><mn>3</mn><mi>x</mi></msup></mrow></math></span>)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa48">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p99"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1553" display="inline"><mrow><msup><mn>2</mn><mrow><mn>2</mn><mi>x</mi></mrow></msup><mo>+</mo><msup><mn>2</mn><mi>x</mi></msup><mo>−</mo><mn>20</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa49">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p101"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1554" display="inline"><mrow><msup><mrow><mn>10</mn></mrow><mrow><mn>2</mn><mi>x</mi></mrow></msup><mo>+</mo><msup><mrow><mn>10</mn></mrow><mi>x</mi></msup><mo>−</mo><mn>12</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa50">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p103"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1556" display="inline"><mrow><msup><mrow><mn>10</mn></mrow><mrow><mn>2</mn><mi>x</mi></mrow></msup><mo>−</mo><msup><mrow><mn>10</mn></mrow><mi>x</mi></msup><mo>−</mo><mn>30</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p105"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1558" display="inline"><mrow><msup><mi>e</mi><mrow><mn>2</mn><mi>x</mi></mrow></msup><mo>−</mo><mn>3</mn><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p107"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1560" display="inline"><mrow><msup><mi>e</mi><mrow><mn>2</mn><mi>x</mi></mrow></msup><mo>−</mo><mn>8</mn><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mn>15</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch07_s05_qs01_qd01_qd05" start="53">
<p class="para" id="fwk-redden-ch07_s05_qs01_p109"><strong class="emphasis bold">Use the change of base formula to approximate the following to the nearest hundredth.</strong></p>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p110"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1563" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa54">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p112"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1564" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>3</mn></msub><mtext> </mtext><mn>7</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa55">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch07_m1565" display="block"><mrow><msub><mrow><mi>log</mi></mrow><mn>5</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa56">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch07_m1566" display="block"><mrow><msub><mrow><mi>log</mi></mrow><mn>7</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>5</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p118"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1567" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub><mtext> </mtext><mn>10</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p120"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1568" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msub><mtext> </mtext><mn>30</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa59">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p122"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1569" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><msqrt><mn>5</mn></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa60">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p124"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1570" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mroot><mn>6</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa61">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p126">If left unchecked, a new strain of flu virus can spread from a single person to others very quickly. The number of people affected can be modeled using the formula <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1571" display="inline"><mrow><mi>P</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>e</mi><mrow><mn>0.22</mn><mi>t</mi></mrow></msup></mrow></math></span>, where <em class="emphasis">t</em> represents the number of days the virus is allowed to spread unchecked. Estimate the number of days it will take 1,000 people to become infected.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa62">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p128">The population of a certain small town is growing according to the function <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1572" display="inline"><mrow><mi>P</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mn>12,500</mn><msup><mrow><mrow><mo>(</mo><mrow><mn>1.02</mn></mrow><mo>)</mo></mrow></mrow><mi>t</mi></msup></mrow></math></span>, where <em class="emphasis">t</em> represents time in years since the last census. Use the function to determine number of years it will take the population to grow to 25,000 people.</p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch07_s05_qs01_qd02">
<h3 class="title">Part B: Solving Logarithmic Equations</h3>
<ol class="qandadiv" id="fwk-redden-ch07_s05_qs01_qd02_qd01" start="63">
<p class="para" id="fwk-redden-ch07_s05_qs01_p130"><strong class="emphasis bold">Solve using the one-to-one property of logarithms.</strong></p>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa63">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p131"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1573" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>5</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>log</mi></mrow><mn>5</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa64">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p133"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1574" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>4</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>7</mn><mi>x</mi></mrow><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>log</mi></mrow><mn>4</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>14</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa65">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p135"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1575" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>−</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>6</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa66">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p137"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1577" display="inline"><mrow><mi>ln</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mi>ln</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa67">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p139"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1578" display="inline"><mrow><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>−</mo><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa68">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p141"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1579" display="inline"><mrow><mi>ln</mi><mtext> </mtext><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi></mrow><mo>)</mo></mrow><mo>=</mo><mn>2</mn><mi>ln</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa69">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p143"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1581" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>3</mn></msub><mtext> </mtext><mn>2</mn><mo>+</mo><mn>2</mn><msub><mrow><mi>log</mi></mrow><mn>3</mn></msub><mtext> </mtext><mi>x</mi><mo>=</mo><msub><mrow><mi>log</mi></mrow><mn>3</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>7</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa70">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p145"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1583" display="inline"><mrow><mn>2</mn><mi>log</mi><mtext> </mtext><mi>x</mi><mo>−</mo><mi>log</mi><mtext> </mtext><mn>36</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa71">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p147"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1584" display="inline"><mrow><mi>ln</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>+</mo><mi>ln</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mi>ln</mi><mtext> </mtext><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa72">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p149"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1585" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>5</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>log</mi></mrow><mn>5</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>log</mi></mrow><mn>5</mn></msub><mtext> </mtext><mn>10</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch07_s05_qs01_qd02_qd02" start="73">
<p class="para" id="fwk-redden-ch07_s05_qs01_p151"><strong class="emphasis bold">Solve.</strong></p>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa73">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p152"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1586" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa74">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p154"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1587" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>3</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa75">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p156"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1588" display="inline"><mrow><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>20</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa76">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p158"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1589" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>4</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa77">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p160"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1590" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>3</mn></msub><mtext> </mtext><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa78">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p162"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1591" display="inline"><mrow><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>10</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa79">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p164"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1592" display="inline"><mrow><mi>ln</mi><mtext> </mtext><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa80">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p166"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1594" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>5</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>20</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa81">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p168"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1596" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>9</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa82">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p170"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1597" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa83">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p172"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1598" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>4</mn></msub><mtext> </mtext><mi>x</mi><mo>+</mo><msub><mrow><mi>log</mi></mrow><mn>4</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa84">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p174"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1599" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>6</mn></msub><mtext> </mtext><mi>x</mi><mo>+</mo><msub><mrow><mi>log</mi></mrow><mn>6</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa85">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p176"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1601" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>3</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>−</mo><msub><mrow><mi>log</mi></mrow><mn>3</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa86">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p178"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1602" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>−</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa87">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p180"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1604" display="inline"><mrow><mi>ln</mi><mtext> </mtext><mi>x</mi><mo>−</mo><mi>ln</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa88">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p182"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1606" display="inline"><mrow><mi>ln</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>−</mo><mi>ln</mi><mtext> </mtext><mi>x</mi><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa89">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p184"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1608" display="inline"><mrow><mn>2</mn><msub><mrow><mi>log</mi></mrow><mn>3</mn></msub><mtext> </mtext><mi>x</mi><mo>=</mo><mn>2</mn><mo>+</mo><msub><mrow><mi>log</mi></mrow><mn>3</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>9</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa90">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p186"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1609" display="inline"><mrow><mn>2</mn><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mi>x</mi><mo>=</mo><mn>3</mn><mo>+</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa91">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p188"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1610" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>2</mn><mo>−</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa92">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p190"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1612" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa93">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p192"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1614" display="inline"><mrow><mi>log</mi><mtext> </mtext><mi>x</mi><mo>−</mo><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa94">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p194"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1615" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mi>x</mi></mrow><mo>)</mo></mrow><mo>=</mo><mn>1</mn><mo>+</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch07_s05_qs01_qd02_qd03" start="95">
<p class="para" id="fwk-redden-ch07_s05_qs01_p196"><strong class="emphasis bold">Find the <em class="emphasis">x</em>- and <em class="emphasis">y</em>-intercepts of the given function.</strong></p>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa95">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p197"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1616" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa96">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p199"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1619" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa97">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p201"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1621" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi></mrow><mo>)</mo></mrow><mo>−</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa98">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p203"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1623" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>log</mi></mrow><mn>3</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa99">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p205"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1626" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>ln</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa100">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p207"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1629" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>ln</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch07_s05_qs01_qd02_qd04" start="101">
<p class="para" id="fwk-redden-ch07_s05_qs01_p209"><strong class="emphasis bold">Find the inverse of the following functions.</strong></p>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa101">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p210"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1632" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa102">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p212"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1634" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>4</mn><mo>+</mo><msub><mrow><mi>log</mi></mrow><mn>3</mn></msub><mtext> </mtext><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa103">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p214"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1636" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa104">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p216"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1638" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>ln</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa105">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p218"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1640" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>ln</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mn>9</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa106">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p220"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1642" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>log</mi></mrow><mn>6</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa107">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p222"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1644" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>e</mi><mrow><mn>3</mn><mi>x</mi></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa108">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p224"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1646" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>2</mn><mi>x</mi></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa109">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p226"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1648" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mn>2</mn><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa110">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p228"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1650" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mn>3</mn><mrow><mn>2</mn><mi>x</mi></mrow></msup><mo>+</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa111">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p230"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1652" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mn>10</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></msup><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa112">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p232"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1654" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>e</mi><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch07_s05_qs01_qd02_qd05" start="113">
<p class="para" id="fwk-redden-ch07_s05_qs01_p234"><strong class="emphasis bold">Solve.</strong></p>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa113">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p235"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1656" display="inline"><mrow><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mn>9</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>1</mn><mo>+</mo><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa114">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p237"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1657" display="inline"><mrow><mn>2</mn><mo>+</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>log</mi></mrow><mn>2</mn></msub><mtext> </mtext><mn>13</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa115">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p239"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1659" display="inline"><mrow><msup><mi>e</mi><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>−</mo><msup><mi>e</mi><mrow><mn>3</mn><mi>x</mi></mrow></msup><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa116">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p241"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1660" display="inline"><mrow><msup><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></msup><mo>−</mo><mn>11</mn><mo>=</mo><mn>70</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa117">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p243"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1661" display="inline"><mrow><msup><mn>2</mn><mrow><mn>3</mn><mi>x</mi></mrow></msup><mo>−</mo><mn>5</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa118">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p245"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1663" display="inline"><mrow><msub><mrow><mi>log</mi></mrow><mn>7</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>log</mi></mrow><mn>7</mn></msub><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa119">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p247"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1665" display="inline"><mrow><mi>ln</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>1</mn><mo>=</mo><mi>ln</mi><mtext> </mtext><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa120">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p249"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1667" display="inline"><mrow><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mn>20</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mi>log</mi><mtext> </mtext><mi>x</mi><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa121">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch07_m1669" display="block"><mrow><mfrac><mn>3</mn><mrow><mn>1</mn><mo>+</mo><msup><mi>e</mi><mrow><mn>2</mn><mi>x</mi></mrow></msup></mrow></mfrac><mo>=</mo><mn>2</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa122">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p253"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1671" display="inline"><mrow><mn>2</mn><msup><mi>e</mi><mrow><mo>−</mo><mn>3</mn><mi>x</mi></mrow></msup><mo>=</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa123">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p255"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1673" display="inline"><mrow><mn>2</mn><msup><mi>e</mi><mrow><mn>3</mn><mi>x</mi></mrow></msup><mo>=</mo><msup><mi>e</mi><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa124">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p257"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1675" display="inline"><mrow><mn>2</mn><mi>log</mi><mtext> </mtext><mi>x</mi><mo>+</mo><mi>log</mi><mtext> </mtext><mi>x</mi><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa125">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p259"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1677" display="inline"><mrow><mn>3</mn><mi>log</mi><mtext> </mtext><mi>x</mi><mo>=</mo><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>2</mn><mi>log</mi><mtext> </mtext><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa126">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p261"><span class="inlineequation"><math xml:id="fwk-redden-ch07_m1678" display="inline"><mrow><mn>2</mn><mi>ln</mi><mtext> </mtext><mn>3</mn><mo>+</mo><mi>ln</mi><mtext> </mtext><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mi>ln</mi><mtext> </mtext><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa127">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p263">In chemistry, pH is a measure of acidity and is given by the formula <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1680" display="inline"><mrow><mtext>pH</mtext><mo>=</mo><mo>−</mo><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><msup><mi>H</mi><mo>+</mo></msup></mrow><mo>)</mo></mrow></mrow></math></span>, where <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1681" display="inline"><mrow><msup><mi>H</mi><mo>+</mo></msup></mrow></math></span> is the hydrogen ion concentration (measured in moles of hydrogen per liter of solution.) Determine the hydrogen ion concentration if the pH of a solution is 4.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa128">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p265">The volume of sound, <em class="emphasis">L</em> in decibels (dB), is given by the formula <span class="inlineequation"><math xml:id="fwk-redden-ch07_m1683" display="inline"><mrow><mi>L</mi><mo>=</mo><mn>10</mn><mi>log</mi><mtext> </mtext><mrow><mo>(</mo><mrow><mi>I</mi><mo>/</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>12</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow></math></span> where <em class="emphasis">I</em> represents the intensity of the sound in watts per square meter. Determine the intensity of an alarm that emits 120 dB of sound.</p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch07_s05_qs01_qd02_qd06" start="129">
<h3 class="title">Part C: Discussion Board</h3>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa129">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p267">Research and discuss the history and use of the slide rule.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa130">
<div class="question">
<p class="para" id="fwk-redden-ch07_s05_qs01_p268">Research and discuss real-world applications involving logarithms.</p>
</div>
</li>
</ol>
</div>
<div class="qandaset block" id="fwk-redden-ch07_s05_qs01_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
<ol class="qandadiv">
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa01_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch07_s05_qs01_p03_ans">4</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa02_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch07_s05_qs01_qa03_ans">