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149 lines
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HTML
149 lines
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<title>Logarithmic Function Reference</title>
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<h1 align="center"> Logarithmic Function Reference</h1>
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<p>This is the <a href="../algebra/logarithms.html">Logarithmic</a> Function:</p>
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<p align="center" class="largest">f(x) = log<sub>a</sub>(x)</p>
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<p align="center"><b>a</b> is any value greater than 0, except 1</p>
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<h3>Properties depend on value of "a" </h3>
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<ul>
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<div class="bigul">
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<li>When <b>a=1</b>, the graph is not defined<b></b></li>
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<li>Apart from that there are two cases to look at:</li>
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</ul>
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<table border="0">
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<tr>
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<td><h3 align="center"><b>a</b> between 0 and 1</h3></td>
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<td width="20"> </td>
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<td><h3 align="center"><b>a</b> above 1</h3></td>
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</tr>
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<tr>
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<td align="center"><img src="images/function-logarithm-lt1.gif" alt="logarithm function" width="183" height="250" /></td>
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<td width="20" align="center"> </td>
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<td align="center"><img src="images/function-logarithm-gt1.gif" alt="logarithm function" width="183" height="251" /></td>
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</tr>
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<tr>
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<td><div align="center"><span class="larger">Example: <b>f(x) = log<sub>½</sub>(x)</b></span></div></td>
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<td width="20"> </td>
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<td><div align="center"><span class="larger">Example: <b>f(x) = log<sub>2</sub>(x)</b></span></div></td>
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</tr>
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<tr>
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<td><p>For <b>a</b> between 0 and 1</p>
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<ul>
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<li>As <b>x</b> nears 0, it heads to <b>infinity</b></li>
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<li>As <b>x</b> increases it heads to <b>-infinity</b></li>
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<li>It is a <a href="functions-increasing.html">Strictly Decreasing</a> function</li>
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<li>It has a <a href="../algebra/asymptote.html">Vertical Asymptote</a> along the y-axis (x=0). </li>
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</ul></td>
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<td width="20"> </td>
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<td><p>For <b>a</b> above 1:</p>
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<ul>
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<li>As <b>x</b> nears 0, it heads to <b>-infinity</b></li>
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<li>As <b>x</b> increases it heads to <b>infinity</b></li>
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<li>it is a <a href="functions-increasing.html">Strictly Increasing</a> function</li>
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<li>It has a <a href="../algebra/asymptote.html">Vertical Asymptote</a> along the y-axis (x=0). </li>
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</ul></td>
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</tr>
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</table>
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<p align="center">Plot the graph <a href="../data/function-grapher8db1.html?func1=log(x)/log(a)&xmin=-12&xmax=12&ymin=-8&ymax=8">here</a> (use the "a" slider)</p>
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<h3>In general, the logarithmic function:</h3>
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<div class="bigul">
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<ul>
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<li>is always on the positive side of (and never crosses) the y-axis</li>
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<li>always intersects the x-axis at x=1 ... in other words it passes through <b>(1,0)</b></li>
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<li>equals <b>1</b> when <b>x=a</b>, in other words it passes through <b>(a,1)</b></li>
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<li>is an <a href="injective-surjective-bijective.html">Injective (one-to-one)</a> function</li>
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</ul>
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</div>
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<div class="center80">
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<p>Its Domain is the Positive Real Numbers: <span class="large">(0, +∞)</span></p>
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<p>Its Range is the <a href="../numbers/real-numbers.html">Real Numbers</a>: <img src="../images/symbols/set-r.svg" alt="Real Numbers" style="vertical-align:middle;" /></p>
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</div>
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<h2>Inverse</h2>
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<div class="def">
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<p align="center"><span class="large"> log<sub>a</sub>(x)</span> is the <a href="function-inverse.html">Inverse Function</a> of <span class="large">a<sup>x</sup></span> (the <a href="function-exponential.html">Exponential Function</a>)</p>
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</div>
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<p>So the Logarithmic Function can be "reversed" by the Exponential Function.</p>
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<h2>The Natural Logarithm Function</h2>
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<p>This is the <b>"Natural"</b> Logarithm Function:</p>
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<p align="center" class="large">f(x) = log<sub>e</sub>(x)</p>
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<p align="center">Where <b>e</b> is "<a href="../numbers/e-eulers-number.html">Eulers Number</a>" = <b>2.718281828459... etc</b></p>
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<p><b>But it is more common to write it this way:</b></p>
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<p align="center" class="largest">f(x) = ln(x)</p>
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<p align="center" class="beach"><i>"ln" meaning "log, natural"</i></p>
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<p>So when you see <span class="large">ln(x)</span>, just remember it is the logarithmic function with base <b>e</b>: <span class="large">log<sub>e</sub>(x)</span>.</p>
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<p align="center"> </p>
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<p align="center"><img src="images/function-logarithm-e.gif" alt="natural logarithm function" width="196" height="225" /><br />
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<span class="larger">Graph of <b>f(x) = ln(x)</b></span></p>
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<p>At the point <b>(e,1)</b> the slope of the line is <b>1/e</b> and the line is tangent to the curve.</p>
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<p> </p>
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<div class="related"><a href="functions-common.html">Common Functions Reference</a> <a href="../algebra/index.html">Algebra Index</a></div>
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