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<title>Working with Exponents and Logarithms</title>
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<h1 align="center">Working with Exponents and Logarithms</h1>
<h2>What is an Exponent?</h2>
<table border="0" align="center" cellpadding="5">
<tr>
<td><img src="images/exponent-2-3.svg" alt="2 with exponent 3" /></td>
<td>&nbsp;</td>
<td><p>The <a href="../exponent.html">exponent</a> of a number says <b>how many times<br />
</b>to
use
the number in a multiplication.</p>
<p class="Larger">In this example: <b>2<sup>3</sup> = 2 &times; 2 &times; 2 = 8</b></p>
<p align="center" class="Larger"><i>(2 is used 3 times in a multiplication to get 8)</i></p></td>
</tr>
</table>
<h2>What is a Logarithm?</h2>
<p>A <a href="logarithms.html">Logarithm</a> goes the other way.</p>
<p>It asks the question &quot;what exponent produced this?&quot;:</p>
<p align="center"><img src="images/logarithm-question.gif" alt="Logarithm Question" width="140" height="63" /></p>
<p>And answers it like this:</p>
<p align="center"><img src="images/exponent-to-logarithm.gif" width="133" height="125" alt="exponent to logarithm" /></p>
<p>In that example:</p>
<ul>
<li>The Exponent takes <b>2 and 3</b> and gives <b>8</b> <i>(2, used 3 times in a multiplication, makes 8)</i></li>
<li>The Logarithm takes <b>2 and 8</b> and gives <b>3</b> <i>(2 makes 8 when used 3 times in a multiplication)</i></li>
</ul>
<div class="def">
<p>A Logarithm says <b>how many</b> of one number to multiply to get another number</p>
</div>
<p>
So a logarithm actually gives you the <b>exponent as its answer</b>:</p>
<p class="center"><img src="images/logarithm-exponent.svg" alt="logarithm concept" /></p>
<i>(Also see how <span class="center"><a href="exponents-roots-logarithms.html">Exponents, Roots and Logarithms</a></span> are related.)</i>
<h2>Working Together</h2>
<p>Exponents and Logarithms work well together because they &quot;undo&quot; each other (so long as the base &quot;a&quot; is the same):</p>
<p align="center"><img src="images/exponent-vs-logarithm.gif" alt="Exponent vs Logarithm" width="175" height="127" /></p>
<p align="center">They are &quot;<a href="../sets/function-inverse.html">Inverse Functions</a>&quot;</p>
<p>&nbsp;</p>
<p>Doing one, then the other, gets you back to where you started:</p>
<div class="tbl">
<div class="row"><span class="left">Doing <b>a<sup>x</sup></b> then <b>log<sub>a</sub></b> gives you <b>x</b> back again:</span><span class="right"><img src="images/loga-ax.gif" alt="Log a (a^x)" width="111" height="25" style="margin-bottom: -5px;" /></span></div>
<div class="row"><span class="left">Doing <b>log<sub>a</sub></b> then <b>a<sup>x</sup></b> gives you <b>x </b>back again:</span><span class="right"><img src="images/aloga-x.gif" alt="a^(log a (x))" width="98" height="24" style="margin-bottom: -2px;" /></span></div>
</div>
<p>&nbsp;</p>
<p>It is too bad they are written <i>so differently</i> ... it makes things look strange. So it may help to think of <span class="large">a<sup>x</sup></span> as &quot;up&quot; and <span class="large">log<sub>a</sub>(x)</span> as &quot;down&quot;:</p>
<div class="tbl">
<div class="row"><span class="left">going up, then down, returns you back again:</span><span class="right">down(up(x)) = x</span></div>
<div class="row"><span class="left">going down, then up, returns you back again:</span><span class="right">up(down(x)) = x</span></div>
</div>
<p>&nbsp;</p>
<p>Anyway, the important thing is that:</p>
<div class="center80">
<p align="center"><b>The Logarithmic Function is &quot;undone&quot; by the Exponential Function.</b></p>
<p align="center"><i>(and vice versa)</i></p>
</div>
<p>Like in this example: </p>
<div class="example">
<h3>Example, what is <b>x</b> in <b> log<sub>3</sub>(x) = 5</b></h3>
<div class="tbl">
<div class="row"><span class="left">Start with:</span><span class="right">log<sub>3</sub>(x) = 5</span></div>
<p>We want to &quot;undo&quot; the log<sub>3</sub> so we can get &quot;x =&quot;</p>
<div class="row"><span class="left">Use the Exponential Function (on both sides):</span><span class="right"><img src="images/3log3-x-35.gif" alt="3^(log3(x))=3^5" width="104" height="24" /></span></div>
<div class="row"><span class="left">And we know that <img src="images/3log3-x-x.gif" alt="3^(log3(x))=x" width="95" height="22" style="margin-bottom: -2px;" />, so:</span><span class="right">x = 3<sup>5</sup></span></div>
<div class="row"><span class="left">Answer: </span><span class="right">x = 243</span></div>
</div>
</div>
<p>And also:</p>
<div class="example">
<h3>Example: Calculate y in <b>y=log<sub>4</sub>(1/4)</b></h3>
<div class="tbl">
<div class="row"><span class="left">Start with:</span><span class="right">y = log<sub>4</sub>(1/4)</span></div>
<div class="row"><span class="left">Use the Exponential Function on both sides:</span><span class="right"><img src="images/4y-4log4-1d4.gif" alt="4^y=4^( log4(1/4) )" width="108" height="26" /></span></div>
<div class="row"><span class="left">Simplify:</span><span class="right">4<sup>y</sup> = 1/4</span></div>
<p>Now a simple trick: <b>1/4 = 4<sup>&minus;1</sup></b></p>
<div class="row"><span class="left">So:</span><span class="right"><span class="larger">4<b><sup>y</sup></b> = 4<b><sup>&minus;1</sup></b></span></span></div>
<div class="row"><span class="left">And so:</span><span class="right"><span class="larger"><b>y = &minus;1</b></span></span></div>
</div>
</div>
<h2>Properties of Logarithms</h2>
<p>One of the powerful things about Logarithms is that they can <b>turn multiply into add</b>.</p>
<div class="def">
<p align="center" class="larger">log<sub>a</sub>( m &times; n ) = log<sub>a</sub>m + log<sub>a</sub>n</p>
<p align="center"><i>&quot;the log of multiplication is the sum of the logs&quot;</i></p>
</div>
<h3 class="center">Why is that true? See <a href="#footnote">Footnote</a>.</h3>
<p>Using that property and the <a href="exponent-laws.html">Laws of Exponents</a> we get these useful properties:</p>
<table border="0">
<tr>
<td width="300" align="center"><span class="larger">log<sub>a</sub>(m &times; n) = log<sub>a</sub>m + log<sub>a</sub>n</span></td>
<td><i>the log of multiplication is the sum of the logs </i></td>
</tr>
<tr>
<td align="center">&nbsp;</td>
<td>&nbsp;</td>
</tr>
<tr>
<td align="center"><span class="larger">log<sub>a</sub>(m/n) = log<sub>a</sub>m &minus; log<sub>a</sub>n</span></td>
<td><i>the log of division is the difference of the logs</i></td>
</tr>
<tr>
<td align="center">&nbsp;</td>
<td>&nbsp;</td>
</tr>
<tr>
<td align="center"><span class="larger">log<sub>a</sub>(1/n) = &minus;log<sub>a</sub>n</span></td>
<td>this just follows on from the previous &quot;division&quot; rule, because <span class="larger">log<sub>a</sub>(1) = 0</span></td>
</tr>
<tr>
<td align="center">&nbsp;</td>
<td>&nbsp;</td>
</tr>
<tr>
<td align="center"><span class="larger">log<sub>a</sub>(m<sup>r</sup>) = r ( log<sub>a</sub>m</span> )</td>
<td><i>the log of m with an exponent r is r times the log of m</i></td>
</tr>
<tr>
<td align="center">&nbsp;</td>
<td>&nbsp;</td>
</tr>
</table>
<p align="center">Remember: the base &quot;a&quot; is always the same! </p>
<div class="center80">
<p><img src="images/logarithm-book.jpg" alt="book of logarithms" width="91" height="90" style="float:left; padding:8px;" /><b>History:</b> Logarithms were very useful before calculators were invented ... for example, instead of multiplying two large numbers, by using logarithms you could turn it into addition (much easier!)</p>
<p>And there were books full of Logarithm tables to help.</p>
</div>
<p>Let us have some fun using the properties:</p>
<div class="example">
<h3>Example: Simplify <b>log<sub>a</sub>( (x<sup>2</sup>+1)<sup>4</sup>&radic;x )</b></h3>
<div class="tbl">
<div class="row"><span class="left">Start with:</span><span class="right">log<sub>a</sub>( (x<sup>2</sup>+1)<sup>4</sup>&radic;x )</span></div>
<div class="row"><span class="left">Use <b>log<sub>a</sub>(mn) = log<sub>a</sub>m + log<sub>a</sub>n</b> :</span><span class="right">log<sub>a</sub>( (x<sup>2</sup>+1)<sup>4</sup> ) + log<sub>a</sub>( &radic;x )</span></div>
<div class="row"><span class="left"> Use <b>log<sub>a</sub>(m<sup>r</sup>) = r ( log<sub>a</sub>m )</b> : </span><span class="right">4 log<sub>a</sub>(x<sup>2</sup>+1) + log<sub>a</sub>( &radic;x )</span></div>
<div class="row"><span class="left">Also <b>&radic;x = x<sup>&frac12;</sup></b> :</span><span class="right">4 log<sub>a</sub>(x<sup>2</sup>+1) + log<sub>a</sub>( x<sup>&frac12;</sup> )</span></div>
<div class="row"><span class="left">Use <b>log<sub>a</sub>(m<sup>r</sup>) = r ( log<sub>a</sub>m ) </b>again: </span><span class="right">4 log<sub>a</sub>(x<sup>2</sup>+1) + &frac12; log<sub>a</sub>(x)</span></div>
</div>
<p>That is as far as we can simplify it ... we can't do anything with <span class="larger">log<sub>a</sub>(x<sup>2</sup>+1)</span>.</p>
<p>&nbsp;</p>
<p align="center">Answer: <span class="large">4 log<sub>a</sub>(x<sup>2</sup>+1) + &frac12; log<sub>a</sub>(x)</span></p>
</div>
<div class="def">
<p>Note: there is no rule for handling <span class="larger">log<sub>a</sub>(m+n) </span>or<span class="larger"> log<sub>a</sub>(m&minus;n)</span></p>
</div>
<p>&nbsp;</p>
<p>We can also apply the logarithm rules "backwards" to combine logarithms: </p>
<div class="example">
<h3>Example: Turn this into one logarithm: <b>log<sub>a</sub>(5) + </b><b>log<sub>a</sub>(x) </b>&minus; <b>log<sub>a</sub>(2) </b></h3>
<div class="tbl">
<div class="row"><span class="left">Start with:</span><span class="right">log<sub>a</sub>(5) + log<sub>a</sub>(x) &minus; log<sub>a</sub>(2)</span></div>
<div class="row"><span class="left">Use <b>log<sub>a</sub>(mn) = log<sub>a</sub>m + log<sub>a</sub>n</b> :</span><span class="right">log<sub>a</sub>(5x) &minus; log<sub>a</sub>(2)</span></div>
<div class="row"><span class="left"> Use <b>log<sub>a</sub>(m/n) = log<sub>a</sub>m &minus; log<sub>a</sub>n</b> : </span><span class="right">log<sub>a</sub>(5x/2)</span></div>
</div>
<p>&nbsp;</p>
<p align="center">Answer: <span class="large">log<sub>a</sub>(5x/2)</span></p>
</div>
<h2>The Natural Logarithm and Natural Exponential Functions</h2>
<p>When the base is <b>e</b> (&quot;<a href="../numbers/e-eulers-number.html">Euler's Number</a>&quot; = <b>2.718281828459</b>...) we get:</p>
<div class="bigul"> <ul>
<li>The Natural Logarithm <b>log<sub>e</sub>(x)</b> which is more commonly written <span class="largest">ln(x)</span></li>
<li>The Natural Exponential Function <span class="largest">e<sup>x</sup></span></li>
</ul></div>
<p>And the same idea that one can &quot;undo&quot; the other is still true:</p>
<p align="center" class="large">ln(e<sup>x</sup>) = x</p>
<p align="center" class="large">e<sup>(ln x)</sup> = x</p>
<p>And here are their graphs:</p><table border="0" align="center">
<tr align="center">
<td><p class="larger">Natural Logarithm</p></td>
<td width="40">&nbsp;</td>
<td><p class="larger">Natural Exponential Function</p></td>
</tr>
<tr align="center">
<td><img src="../sets/images/function-logarithm-e.gif" alt="natural logarithm function" width="196" height="225" /></td>
<td width="40">&nbsp;</td>
<td><img src="../sets/images/function-exponential-e-pts.svg" alt="natural exponential function" /></td>
</tr>
<tr align="center">
<td><span class="larger">Graph of <b>f(x) = ln(x)</b></span></td>
<td width="40">&nbsp;</td>
<td><div align="center"><span class="larger">Graph of <b>f(x) = e<sup>x</sup></b></span></div></td>
</tr>
<tr align="center">
<td><p>Passes through <b>(1,0)</b> and <b>(e,1)</b></p> </td>
<td width="40">&nbsp;</td>
<td><p>Passes through <b>(0,1)</b> and <b>(1,e)</b></p> </td>
</tr>
</table>
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/lnx-vs-expx.gif" width="180" height="122" alt="ln(x) vs e^x" /></p>
<p>They are the <b>same curve</b> with x-axis and y-axis <b>flipped</b>. </p>
<p>Which is another thing to show you they are inverse functions.</p>
<table border="0">
<tr>
<td><img src="../money/images/calculator-ln.gif" width="107" height="87" alt="calculator ln button" /></td>
<td>&nbsp;</td>
<td><p>On a calculator the Natural Logarithm is the &quot;ln&quot; button.</p></td>
</tr>
</table>
<p>Always try to use Natural Logarithms and the Natural Exponential Function whenever possible.</p>
<h2>The Common Logarithm </h2>
<p>When the base is <b>10</b> you get:</p>
<ul>
<div class="bigul">
<li>The Common Logarithm<b> log<sub>10</sub>(x)</b>, which is sometimes written as <span class="largest">log(x)</span></li>
</div>
</ul>
<p>Engineers love to use it, but it is not used much in mathematics.</p>
<table border="0">
<tr>
<td><img src="../money/images/calculator-log.gif" alt="calculator log button" width="114" height="91" /></td>
<td>&nbsp;</td>
<td><p>On a calculator the Common Logarithm is the &quot;log&quot; button.</p>
<p>It is handy because it tells you how &quot;big&quot; the number is in decimal (how many times you need to use 10 in a multiplication).</p></td>
</tr>
</table>
<div class="example">
<h3>Example: Calculate log<sub>10</sub> 100</h3>
<p>Well, 10 &times; 10 = 100, so when 10 is used <b>2</b> times in a multiplication you get 100:</p>
<p align="center"><span class="large">log<sub>10</sub> 100 = 2</span></p>
<p>Likewise log<sub>10</sub> 1,000 = 3, log<sub>10</sub> 10,000 = 4, and so on.</p>
</div>
<div class="example">
<h3>Example: Calculate log<sub>10</sub> 369</h3>
<p>OK, best to use my calculator's &quot;log&quot; button:</p>
<p align="center"><span class="large">log<sub>10</sub> 369 = 2.567...</span></p>
</div>
<h2>Changing the Base</h2>
<p>What if we want to change the base of a logarithm?</p>
<p>Easy! Just use this formula:</p>
<p align="center"><img src="images/log-chg-base.gif" alt="Log Change Base" width="203" height="146" /></p>
<p align="center">&quot;x goes up, a goes down&quot;</p>
<p>Or another way to think of it is that <b>log<sub>b</sub> a</b> is like a &quot;conversion factor&quot; (same formula as above):</p>
<div class="def">
<p align="center"><span class="larger">log<sub>a</sub> x = log<sub>b</sub> x /<b> log<sub>b</sub> a</b></span></p>
</div>
<p>So now we can convert from any base to any other base. </p>
<p>Another useful property is:</p>
<div class="def">
<p align="center"><span class="larger">log<sub>a</sub> x = 1 / log<sub>x</sub> a</span></p>
</div>
<p>See how &quot;x&quot; and &quot;a&quot; swap positions? </p>
<div class="example">
<h3>Example: Calculate 1 / log<sub>8</sub> 2</h3>
<p align="center"><span class="large">1 / log<sub>8</sub> 2 = log<sub>2</sub> 8</span></p>
<p>And 2 &times; 2 &times; 2 = 8, so when 2 is used <b>3</b> times in a multiplication you get 8:</p>
<p align="center"><span class="large">1 / log<sub>8</sub> 2 = log<sub>2</sub> 8 = 3</span></p>
</div>
<p>&nbsp;</p>
<p>But we use the Natural Logarithm more often, so this is worth remembering:</p>
<div class="def">
<p align="center"><span class="large">log<sub>a</sub> x = ln x / ln a</span></p>
</div>
<p>&nbsp;</p>
<div class="example">
<h3>Example: Calculate log<sub>4</sub> 22</h3>
<table border="0">
<tr>
<td><img src="../money/images/calculator-ln.gif" width="107" height="87" alt="calculator ln button" /></td>
<td><p>My calculator doesn't have a &quot;<b>log<sub>4</sub></b>&quot; button ...</p>
<p>... but it does have an &quot;<b>ln</b>&quot; button, so we can use that:</p></td>
</tr>
</table>
<div class="tbl">
<div class="row"><span class="left">log<sub>4</sub> 22 =</span><span class="right"> ln 22 / ln 4 </span></div>
<div class="row"><span class="left"> =</span><span class="right"> 3.09.../1.39... </span></div>
<div class="row"><span class="left"> =</span><span class="right"> 2.23 <i>(to 2 decimal places)</i></span></div>
</div>
<p align="center">&nbsp;</p>
<p>What does this answer mean? It means that 4 with an exponent of 2.23 equals 22. So we can check that answer:</p>
<p align="center"><span class="larger">Check: 4<sup>2.23</sup> = 22.01</span> (close enough!)</p>
</div>
<p>Here is another example:</p>
<div class="example">
<h3>Example: Calculate log<sub>5</sub> 125</h3>
<div class="tbl">
<div class="row"><span class="left">log<sub>5</sub> 125 =</span><span class="right"> ln 125 / ln 5 </span></div>
<div class="row"><span class="left"> =</span><span class="right"> 4.83.../1.61... </span></div>
<div class="row"><span class="left"> =</span><span class="right">3 <i> (exactly)</i> </span></span></div>
</div>
<p align="center">&nbsp;</p>
<p>I happen to know that 5 &times; 5 &times; 5 = 125, (5 is used <b>3</b> times to get 125), so I expected an answer of <b>3</b>, and it worked!</p>
</div>
<h2>Real World Usage</h2>
<p>Here are some uses for Logarithms in the real world:</p>
<h3>Earthquakes</h3>
<p>The magnitude of an earthquake is a Logarithmic scale.</p>
<p>The famous &quot;Richter Scale&quot; uses this formula:</p>
<p align="center" class="large">M = log<sub>10</sub> A + B</p>
<p align="center">Where <b>A</b> is the amplitude (in mm) measured by the Seismograph<br />
and <b>B </b>is a distance correction factor</p>
<p>Nowadays there are more complicated formulas, but they still use a logarithmic scale.</p>
<h3>Sound </h3>
<p>Loudness is measured in Decibels (dB for short):</p>
<p align="center"><span class="large">Loudness in dB = 10 log<sub>10</sub> (p &times; 10<sup>12</sup>)</span></p>
<p align="center">where <b>p</b> is the sound pressure.</p>
<h3>Acidic or Alkaline</h3>
<p> Acidity (or Alkalinity) is measured in pH:</p>
<p align="center"><span class="large">pH = &minus;log<sub>10</sub> [H<sup>+</sup>]</span></p>
<p align="center">where <b>H<sup>+</sup></b> is the molar concentration of dissolved hydrogen ions.<br />
Note: in chemistry [ ] means molar concentration (moles per liter). </p>
<h2>More Examples</h2>
<div class="example">
<h3>Example: Solve 2 log<sub>8</sub> x = log<sub>8</sub> 16</h3>
<div class="tbl">
<div class="row"><span class="left">Start with:</span><span class="right">2 log<sub>8</sub> x = log<sub>8</sub> 16</span></div>
<div class="row"><span class="left">Bring the &quot;2&quot; into the log:</span><span class="right">log<sub>8</sub> x<sup>2</sup> = log<sub>8</sub> 16</span></div>
<div class="row"><span class="left"> Remove the logs (they are same base): </span><span class="right">x<sup>2</sup> = 16</span></div>
<div class="row"><span class="left">Solve:</span><span class="right">x = &minus;4 or +4</span></div>
</div>
<p>But ... but ... but ... you cannot have a log of a negative number!</p>
<p>So the &minus;4 case is not defined.</p>
<p align="center" class="large">Answer: 4</p>
<p>Check: use your calculator to see if this is the right answer ... also try the &quot;&minus;4&quot; case.</p>
</div>
<div class="example">
<h3>Example: Solve e<sup><span class="larger">&minus;</span>w</sup> = e<sup>2w+6</sup></h3>
<div class="tbl">
<div class="row"><span class="left">Start with:</span><span class="right">e<sup>&minus;w</sup> = e<sup>2w+6</sup></span></div>
<div class="row"><span class="left">Apply <b>ln</b> to both sides:</span><span class="right">ln(e<sup>&minus;w</sup>) = ln(e<sup>2w+6</sup>)</span></div>
<div class="row"><span class="left">And <b>ln(e<sup>w</sup></b><b>)=w</b>: </span><span class="right">&minus;w = 2w+6</span></div>
<div class="row"><span class="left">Simplify:</span><span class="right">&minus;3w = 6</span></div>
<div class="row"><span class="left">Solve:</span><span class="right">w = 6/&minus;3 = &minus;2</span></div>
</div>
<p align="center" class="large">Answer: w = <span class="larger">&minus;</span>2</p>
<p>Check: e<sup>&minus;(&minus;2)</sup>= e<sup>2</sup> and e<sup>2(&minus;2)+6</sup>=e<sup>2</sup></p>
</div>
<p>&nbsp;</p>
<div class="def">
<h3><a name="footnote"></a>Footnote: Why does <b>log(m &times; n) = log(m) + log(n)</b> ?</h3>
<p>To see <b>why</b>, we will use <img src="images/aloga-x.gif" alt="a^(log a (x))" width="98" height="24" style="margin-bottom: -2px;" /> and <img src="images/loga-ax.gif" alt="Log a (a^x)" width="111" height="25" style="margin-bottom: -7px;" />:</p>
<table border="0" align="center">
<tr>
<td align="center">First, make <b>m</b> and <b>n</b> into &quot;exponents of logarithms&quot;:</td>
<td>&nbsp;</td>
</tr>
<tr>
<td><img src="images/log-rule-explained.gif" alt="Log Producr Rule" width="460" height="163" /></td>
<td valign="bottom"><p>&nbsp;</p>
<p>Then&nbsp;use&nbsp;one of the <a href="exponent-laws.html">Laws of Exponents</a></p>
<p>Finally undo the exponents.</p></td>
</tr>
</table>
<p>It is one of those clever things we do in mathematics which can be described as <i>&quot;we can't do it here, so let's go over <b>there</b>, then do it, then come back&quot;</i> </p>
</div>
<p>&nbsp;</p>
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<div class="related"><a href="../exponent.html">Exponents</a> <a href="logarithms.html">Logarithms</a> <a href="index-2.html">Algebra 2 Index</a></div>
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