e39465ad2f
new file: Files/flashplayer_32_sa.exe
new file: favicon.ico
new file: globe.gif
new file: imgs/download.png
new file: imgs/zuck.jpg
new file: index.html
new file: other.ico
new file: script.js
new file: site.webmanifest
new file: sitemap.html
new file: styles/backround.css
new file: styles/border.css
new file: styles/fonts/Titillium_Web/OFL.txt
new file: styles/fonts/Titillium_Web/TitilliumWeb-Black.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Bold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-BoldItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLight.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Italic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Light.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-LightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Regular.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBoldItalic.ttf
new file: styles/fonts/webfontkit-20221027-163353/generator_config.txt
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-demo.html
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff2
new file: styles/fonts/webfontkit-20221027-165950/generator_config.txt
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-demo.html
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff2
new file: styles/style.css
new file: tools/2048/.gitignore
new file: tools/2048/.jshintrc
new file: tools/2048/CONTRIBUTING.md
new file: tools/2048/LICENSE.txt
new file: tools/2048/README.md
new file: tools/2048/Rakefile
new file: tools/2048/favicon.ico
new file: tools/2048/index.html
new file: tools/2048/js/animframe_polyfill.js
new file: tools/2048/js/application.js
new file: tools/2048/js/bind_polyfill.js
new file: tools/2048/js/classlist_polyfill.js
new file: tools/2048/js/game_manager.js
new file: tools/2048/js/grid.js
new file: tools/2048/js/html_actuator.js
new file: tools/2048/js/keyboard_input_manager.js
new file: tools/2048/js/local_storage_manager.js
new file: tools/2048/js/tile.js
new file: tools/2048/meta/apple-touch-icon.png
new file: tools/webretro/cores/neocd_libretro.js
new file: tools/webretro/cores/neocd_libretro.wasm
new file: tools/webretro/cores/nestopia_libretro.js
new file: tools/webretro/cores/nestopia_libretro.wasm
new file: tools/webretro/cores/o2em_libretro.js
new file: tools/webretro/cores/o2em_libretro.wasm
new file: tools/webretro/cores/opera_libretro.js
new file: tools/webretro/cores/opera_libretro.wasm
397 lines
16 KiB
HTML
397 lines
16 KiB
HTML
<!DOCTYPE html>
|
||
<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
|
||
|
||
<!-- Mirrored from www.mathsisfun.com/algebra/logarithms.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:35:44 GMT -->
|
||
<head>
|
||
<meta http-equiv="content-type" content="text/html; charset=UTF-8">
|
||
|
||
<!-- #BeginEditable "doctitle" -->
|
||
<title>Introduction to Logarithms</title>
|
||
<meta name="Description" content="Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents."><!-- #EndEditable -->
|
||
<meta name="keywords" content="math, maths, mathematics, school, homework, education">
|
||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
|
||
<meta name="HandheldFriendly" content="true">
|
||
<meta name="referrer" content="always">
|
||
<link rel="preload" href="../images/style/font-champ-bold.ttf" as="font" type="font/ttf" crossorigin="">
|
||
<link rel="preload" href="../style4.css" as="style">
|
||
<link rel="preload" href="../main4.js" as="script">
|
||
<link rel="stylesheet" href="../style4.css">
|
||
<script src="../main4.js" defer="defer"></script>
|
||
<!-- Global site tag (gtag.js) - Google Analytics -->
|
||
<script async="" src="https://www.googletagmanager.com/gtag/js?id=UA-29771508-1"></script>
|
||
<script>
|
||
window.dataLayer = window.dataLayer || [];
|
||
function gtag(){dataLayer.push(arguments);}
|
||
gtag('js', new Date());
|
||
gtag('config', 'UA-29771508-1');
|
||
</script>
|
||
</head>
|
||
|
||
<body id="bodybg">
|
||
|
||
<div id="stt"></div>
|
||
<div id="adTop"></div>
|
||
<header>
|
||
<div id="hdr"></div>
|
||
<div id="tran"></div>
|
||
<div id="adHide"></div>
|
||
<div id="cookOK"></div>
|
||
</header>
|
||
|
||
<div class="mid">
|
||
|
||
<nav>
|
||
<div id="menuWide" class="menu"></div>
|
||
<div id="logo"><a href="../index.html"><img src="../images/style/logo.svg" alt="Math is Fun"></a></div>
|
||
|
||
<div id="search" role="search"></div>
|
||
<div id="linkto"></div>
|
||
|
||
<div id="menuSlim" class="menu"></div>
|
||
<div id="menuTiny" class="menu"></div>
|
||
</nav>
|
||
|
||
<div id="extra"></div>
|
||
|
||
<article id="content" role="main">
|
||
|
||
<!-- #BeginEditable "Body" -->
|
||
|
||
|
||
<h1 class="center">Introduction to Logarithms</h1>
|
||
|
||
<p>In its simplest form, a logarithm answers the question:</p>
|
||
<div class="def">
|
||
<p class="center"><b>How many of <i>one number</i> multiply together to make <i>another number?</i></b></p>
|
||
</div>
|
||
|
||
<div class="example">
|
||
<p class="large">Example: How many <span class="larger">2</span>s multiply together to make <span class="larger">8</span>?</p>
|
||
<p class="center">Answer: <b>2 × 2 × 2 = 8</b>, so we had to multiply <span class="large">3</span> of the <b>2</b>s to get <b>8</b></p>
|
||
<p class="large">So the logarithm is <b>3</b></p>
|
||
</div>
|
||
|
||
<h3>How to Write it</h3>
|
||
<p>We write it like this:</p>
|
||
<p class="center"><span class="larger">log<sub>2</sub>(8) = 3</span></p>
|
||
<p> So these two things are the same:</p>
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td><img src="images/logarithm-concept.svg" alt="logarithm concept 2x2x2=8 same as log_2(8)=3" height="86" width="424"></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<div class="center80">
|
||
<p>The number we multiply is called the "base", so we can say:</p>
|
||
<ul>
|
||
<li>"the logarithm of 8 with base 2 is 3"</li>
|
||
<li>or "log base 2 of 8 is 3"</li>
|
||
<li>or "the base-2 log of 8 is 3"</li>
|
||
</ul>
|
||
</div>
|
||
|
||
<h3>Notice we are dealing with three numbers:</h3>
|
||
<ul>
|
||
<li>the <b>base</b>: the number we are multiplying (a "2" in the example above)</li>
|
||
<li>how often to use it in a multiplication (3 times, which is the <b>logarithm</b>)</li>
|
||
<li>The number we want to get (an "8")</li>
|
||
</ul>
|
||
|
||
|
||
<h2>More Examples</h2>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: What is <span class="large">log<sub>5</sub>(625) </span>... ?</h3>
|
||
<p>We are asking "how many 5s need to be multiplied together to get 625?"</p>
|
||
<p><b>5 × 5 × 5 × 5 = 625</b>, so we need <span class="large">4</span> of the 5s</p>
|
||
<p>Answer: <span class="large">log<sub>5</sub>(625) = 4</span></p>
|
||
</div>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: What is <span class="large">log<sub>2</sub>(64) </span>... ?</h3>
|
||
<p>We are asking "how many 2s need to be multiplied together to get 64?"</p>
|
||
<p><b>2 × 2 × 2 × 2 × 2 × 2 = 64</b>, so we need <span class="large">6</span> of the 2s</p>
|
||
<p>Answer: <span class="large">log<sub>2</sub>(64) = 6</span></p>
|
||
</div>
|
||
|
||
|
||
<h2>Exponents</h2>
|
||
|
||
<p>Exponents and Logarithms are related, let's find out how ...</p>
|
||
<div class="simple">
|
||
|
||
<table align="center" cellpadding="5" border="0">
|
||
<tbody>
|
||
<tr>
|
||
<td><img src="images/exponent-2-3.svg" alt="2 cubed" height="132" width="143"></td>
|
||
<td>
|
||
<p>The <b>exponent</b> says <b>how many times</b> to use the number in a multiplication.</p>
|
||
<p class="larger">In this example: <b>2<sup>3</sup> = 2 × 2 × 2 = 8</b></p>
|
||
<p class="center large"><i>(2 is used 3 times in a multiplication to get 8)</i></p></td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p>So a logarithm answers a question like this:</p>
|
||
<p class="center"><img src="images/logarithm-question.svg" alt="2 with what exponent = 8"></p>
|
||
|
||
<p>In this way:</p>
|
||
<p class="center"><img src="images/exponent-to-logarithm.svg" alt="2^3=8 becomes log_2(8)=3"></p>
|
||
<p class="center"><b>The logarithm tells us what the exponent is!</b></p>
|
||
<p>In that example the "base" is 2 and the "exponent" is 3:</p>
|
||
<p class="center"><img src="images/logarithm-exponent.svg" alt="2^3=8 becomes log_2(8)=3" height="99" width="322"></p>
|
||
<p>So the logarithm answers the question:</p>
|
||
<div class="def">
|
||
<p class="center larger"><b>What exponent do we need</b> <i><br>
|
||
(for one number to become another number)</i><b> ?</b></p>
|
||
</div>
|
||
<p>The <b>general</b> case is:</p>
|
||
<p class="center">
|
||
<img src="images/exponent-to-logarithm-gen.svg" alt="a^x=y becomes log_a(y)=x"> </p>
|
||
<p></p>
|
||
|
||
<div class="example">
|
||
<p>Example: What is <span class="large">log<sub>10</sub>(100) </span>... ?</p>
|
||
<p class="center"><span class="large">10<sup>2</sup> = 100</span></p>
|
||
<p>So an exponent of <span class="large">2</span> is needed to make 10 into 100, and:</p>
|
||
<p class="center"><span class="large">log<sub>10</sub>(100) = 2</span></p>
|
||
</div>
|
||
|
||
<div class="example">
|
||
<p>Example: What is <span class="large">log<sub>3</sub>(81) </span>... ?</p>
|
||
<p class="center"><span class="large">3<sup>4</sup> = 81</span></p>
|
||
<p>So an exponent of <span class="large">4</span> is needed to make 3 into 81, and:</p>
|
||
<p class="center"><span class="large">log<sub>3</sub>(81) = 4</span></p>
|
||
</div>
|
||
<p> </p>
|
||
|
||
|
||
<h2>Common Logarithms: Base 10</h2>
|
||
|
||
<p>Sometimes a logarithm is written <b>without</b> a base, like this:</p>
|
||
<p class="center large">log(100)</p>
|
||
|
||
<p>This <i><b>usually</b></i> means that the base is really <span class="large">10</span>.</p>
|
||
<p style="float:left; margin: 0 10px 5px 0;"><img src="../money/images/calculator-log.gif" alt="log" height="91" width="114"></p>
|
||
<p>It is called a "common logarithm". Engineers love to use it.</p>
|
||
<p>On a calculator it is the "log" button.</p>
|
||
<p>It is how many times we need to use 10 in a multiplication, to get our desired number.</p>
|
||
|
||
<div class="example">
|
||
<p class="larger">Example: <b>log(1000) = <span class="large">log<sub>10</sub>(1000) = 3</span></b></p>
|
||
</div><p> </p>
|
||
|
||
|
||
<h2>Natural Logarithms: Base "e"</h2>
|
||
|
||
<p>Another base that is often used is <a href="../numbers/e-eulers-number.html">e (Euler's Number)</a> which is about 2.71828.</p>
|
||
<p style="float:left; margin: 0 10px 5px 0;"><img src="../money/images/calculator-ln.gif" alt="calculator ln button" height="87" width="107"></p>
|
||
<p>This is called a "natural logarithm". Mathematicians use this one a lot.</p>
|
||
<p>On a calculator it is the "ln" button.</p>
|
||
<p>It is how many times we need to use "e" in a multiplication, to get our desired number.</p>
|
||
|
||
<div class="example">
|
||
<p class="larger">Example: <b>ln(7.389) = <span class="large">log<sub>e</sub>(</span>7.389<span class="large">) ≈ 2</span></b></p>
|
||
<p>Because <b>2.71828<sup>2</sup> ≈ 7.389</b></p>
|
||
</div><br>
|
||
<h2>But Sometimes There Is Confusion ... !</h2>
|
||
|
||
<p>Mathematicians may use "log" (instead of "ln") to mean the natural logarithm. This can lead to confusion:</p>
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr style="text-align:center;">
|
||
<th width="140">Example</th>
|
||
<th width="140">Engineer<br>
|
||
Thinks</th>
|
||
<th width="140">Mathematician<br>
|
||
Thinks</th>
|
||
<th> </th>
|
||
</tr>
|
||
<tr style="color:#ff6666" align="center">
|
||
<td>log(50)</td>
|
||
<td>log<sub>10</sub>(50)</td>
|
||
<td>log<sub>e</sub>(50)</td>
|
||
<td>confusion</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>ln(50)</td>
|
||
<td>log<sub>e</sub>(50)</td>
|
||
<td>log<sub>e</sub>(50)</td>
|
||
<td>no confusion</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>log<sub>10</sub>(50)</td>
|
||
<td>log<sub>10</sub>(50)</td>
|
||
<td>log<sub>10</sub>(50)</td>
|
||
<td>no confusion</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>So, be careful when you read "log" that you know what base they mean!</p>
|
||
<p> </p>
|
||
|
||
|
||
<h2>Logarithms Can Have Decimals</h2>
|
||
|
||
<p>All of our examples have used whole number logarithms (like 2 or 3), but logarithms can have decimal values like 2.5, or 6.081, etc.</p>
|
||
|
||
<div class="example">
|
||
<p class="larger">Example: what is <b><span class="large">log<sub>10</sub>(26) ... ?</span></b></p>
|
||
|
||
<table width="100%" border="0">
|
||
<tbody>
|
||
<tr>
|
||
<td width="16%"><img src="../money/images/calculator-log.gif" alt="log" height="91" width="114"></td>
|
||
<td width="84%">
|
||
<p>Get your calculator, type in <b>26</b> and press <b>log</b></p>
|
||
<p>Answer is: <b>1.41497...</b></p></td>
|
||
</tr>
|
||
</tbody></table>
|
||
|
||
<p>The logarithm is saying that 10<sup>1.41497...</sup> = 26<br>
|
||
(10 with an exponent of <b>1.41497...</b> equals 26)</p>
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td>
|
||
<p>This is what it looks like on a graph:</p>
|
||
<p>See how nice and smooth the line is.</p></td>
|
||
<td> </td>
|
||
<td><img src="images/log-10-26.svg" alt="log 10 of 26" height="" width=""></td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p>Read <a href="logarithms-decimals.html">Logarithms Can Have Decimals</a> to find out more.</p>
|
||
|
||
|
||
<h2>Negative Logarithms</h2>
|
||
|
||
<table width="100%" border="0">
|
||
<tbody>
|
||
<tr>
|
||
<td align="center" width="12%"><span class="huge">−</span></td>
|
||
<td width="88%">Negative? But logarithms deal with multiplying.<br>
|
||
What is the opposite of multiplying? <b>Dividing!</b> </td>
|
||
</tr>
|
||
</tbody></table><br>
|
||
<div class="def">
|
||
<p class="center">A negative logarithm means how many times<b> to
|
||
divide</b> by the number.</p>
|
||
</div>
|
||
<p>We can have just one divide:</p>
|
||
|
||
<div class="example">
|
||
<p>Example: What is <span class="large">log<sub>8</sub>(0.125)</span> ... ?</p>
|
||
<p>Well, <span class="large">1 ÷ 8 = 0.125</span>,</p>
|
||
<p>So <span class="large">log<sub>8</sub>(0.125) = −1</span></p>
|
||
</div>
|
||
<p>Or many divides:</p>
|
||
|
||
<div class="example">
|
||
<p>Example: What is <span class="large">log<sub>5</sub>(0.008)</span> ... ?</p>
|
||
<p><span class="large"><b>1 ÷ 5 ÷ 5 ÷ 5</b> = <b>5<sup>-3</sup></b></span>,</p>
|
||
<p>So <span class="large"> log<sub>5</sub>(0.008) = −3</span></p>
|
||
</div>
|
||
|
||
|
||
<h2>It All Makes Sense</h2>
|
||
|
||
<p>Multiplying and Dividing are all part of the same simple pattern.</p>
|
||
<p>Let us look at some Base-10 logarithms as an example:</p>
|
||
<div class="simple">
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr style="text-align:center;">
|
||
<td bgcolor="#FFFFFF"> </td>
|
||
<th>Number</th>
|
||
<th>How Many 10s</th>
|
||
<th colspan="2" align="right"> Base-10 Logarithm</th>
|
||
</tr>
|
||
<tr>
|
||
<td rowspan="9" align="center" bgcolor="#FFFFFF"><img src="images/larger-smaller-10.svg" alt="10 times larger / smaller" height="199" width="69"></td>
|
||
<td style="text-align:center;">.. etc..</td>
|
||
<td style="text-align:center;"> </td>
|
||
<td style="text-align:right;"> </td>
|
||
<td style="text-align:center;"> </td>
|
||
</tr>
|
||
<tr>
|
||
<td class="large">1000</td>
|
||
<td>1 × 10 × 10 × 10</td>
|
||
<td style="text-align:right;">log<sub>10</sub>(1000)</td>
|
||
<td class="large" align="center">= 3</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="large">100</td>
|
||
<td>1 × 10 × 10</td>
|
||
<td style="text-align:right;">log<sub>10</sub>(100)</td>
|
||
<td class="large" align="center">= 2</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="large">10</td>
|
||
<td>1 × 10</td>
|
||
<td style="text-align:right;">log<sub>10</sub>(10)</td>
|
||
<td class="large" align="center">= 1</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="large">1</td>
|
||
<td>1</td>
|
||
<td style="text-align:right;">log<sub>10</sub>(1)</td>
|
||
<td class="large" align="center">= 0</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="large">0.1</td>
|
||
<td>1 ÷ 10</td>
|
||
<td style="text-align:right;">log<sub>10</sub>(0.1)</td>
|
||
<td class="large" align="center">= −1</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="large">0.01</td>
|
||
<td>1 ÷ 10 ÷ 10</td>
|
||
<td style="text-align:right;">log<sub>10</sub>(0.01)</td>
|
||
<td class="large" align="center">= −2</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="large">0.001</td>
|
||
<td>1 ÷ 10 ÷ 10 ÷ 10</td>
|
||
<td style="text-align:right;">log<sub>10</sub>(0.001)</td>
|
||
<td class="large" align="center">= −3</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center;">.. etc..</td>
|
||
<td style="text-align:center;"> </td>
|
||
<td style="text-align:right;"> </td>
|
||
<td style="text-align:center;"> </td>
|
||
</tr>
|
||
</tbody></table> </div>
|
||
<p>Looking at that table, see how positive, zero or negative logarithms are really part of the same (fairly simple) pattern.</p>
|
||
<p> </p>
|
||
|
||
|
||
<h2>The Word</h2>
|
||
|
||
<div class="words">
|
||
<p>"Logarithm" is a word made up by Scottish mathematician John Napier (1550-1617), from the Greek word <i>logos</i> meaning "proportion, ratio or word" and <i>arithmos</i> meaning "number", ... which together makes "ratio-number" !</p>
|
||
</div>
|
||
<div class="questions">340, 341, 2384, 2385, 2386, 2387, 3180, 3181, 2388, 2389</div>
|
||
|
||
<div class="related">
|
||
<a href="exponents-roots-logarithms.html">Exponents, Roots and Logarithms</a>
|
||
<a href="exponents-logarithms.html">Working with Exponents and Logarithms</a>
|
||
<a href="index.html">Algebra Index</a>
|
||
</div>
|
||
<!-- #EndEditable -->
|
||
|
||
</article>
|
||
|
||
<div id="adend" class="centerfull noprint"></div>
|
||
<footer id="footer" class="centerfull noprint"></footer>
|
||
<div id="copyrt">Copyright © 2022 Rod Pierce</div>
|
||
|
||
</div>
|
||
</body><!-- #EndTemplate -->
|
||
<!-- Mirrored from www.mathsisfun.com/algebra/logarithms.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:35:46 GMT -->
|
||
</html> |