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188 lines
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HTML
188 lines
8.0 KiB
HTML
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Logarithms Can Have Decimals</h1>
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<p> </p>
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<p>On <a href="logarithms.html">Introduction to Logarithms</a> we saw that a logarithm answers questions like this:</p>
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<div class="example">
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<p class="larger">How many <span class="large">2</span>s do we multiply to get <span class="large">8</span>?</p>
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<p class="center">Answer: <b>2 × 2 × 2 = 8</b>, so we needed to multiply <span class="large">3</span> of the <b>2</b>s to get <b>8</b></p>
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<p class="larger">So the logarithm is 3</p>
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</div>
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<p>And we write "the number of 2s we multiply to get 8 is <b>3</b>" as</p>
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<p class="center"><span class="large">log<sub>2</sub>(8) = 3</span></p>
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So these two things are the same:<p class="center"><img src="images/logarithm-concept.svg" alt="logarithm concept 2x2x2=8 same as log_2(8)=3" /></p>
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<p> </p>
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<div class="example">
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<h3>Example: What is <span class="large">log<sub>10</sub>(100) </span>... ?</h3>
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<p class="center"><span class="large">10 × 10 = 100</span></p>
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<p>Multiplying <span class="large">2</span> 10s together makes 100, so:</p>
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<p class="center"><span class="large">log<sub>10</sub>(100) = 2</span></p>
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<p> </p>
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<p>Note: using exponents it is: <b>10<sup>2</sup> = 100</b></p>
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</div>
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<p>But now we ask a new question:</p>
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<div class="example">
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<h3>Example: What is <span class="large">log<sub>10</sub>(300) </span>... ?</h3>
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<p class="center"><span class="large">10 × 10 = 100</span></p>
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<p class="center"><span class="large">10 × 10 × 10 = 1000</span></p>
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<p>Oh no! We are either too low or too high.</p>
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</div>
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<p>So multiplying <b>two</b> 10s is not enough, but multiplying <b>three</b> 10s is too many ...</p>
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<p class="center">... but what about <b>two and a half</b> ... ?</p>
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<p> </p>
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<h2>Half a Multiply ...</h2>
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<div class="center80">
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<p>How can we do <b> half a multiply</b>?</p>
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<p> </p>
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<p>Well, <b>half a multiply</b> is something we need to do <b>twice</b> to make a <b>whole multiply</b>.</p>
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<p> </p>
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<p>And that is <a href="../square-root.html">square root</a> !</p>
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<p class="center larger">√10 × √10 = 10</p>
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</div>
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<p>Multiplying by a square root is like doing half a multiply.</p>
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<p>So let us try that:</p>
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<div class="example">
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<h3>Example: <span class="large">log<sub>10</sub>(300) </span>(continued)</h3>
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<p>Try using 10 in a multiplication <b>two and a half times</b>:</p>
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<p class="center"><span class="large">10 × 10 × √10 <br>
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=
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10 × 10 × 3.16... <br>
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= 316....</span></p>
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<p>We are close to 300, so we could say:</p>
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<p class="center"><span class="large">log<sub>10</sub>(300) ≈ 2.5 (approximately)</span></p>
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<p>In other words using 10 in a multiplication two and a half times gets approximately 300.</p>
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<p>(Note: using exponents we can say <b>300 ≈ 10<sup>2.5</sup></b>)</p>
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</div>
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<p>And this is what it looks like on a graph:</p>
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<p class="center"><img src="images/log-10-graph.gif" width="477" height="269" alt="log 10 graph" /></p>
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<p class="center"><span class="larger">2:</span> 10 × 10 = <b>100</b><br>
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<span class="larger">2.5:</span> 10 × 10 × √10 = <b>316....</b><br>
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<span class="larger">3:</span> 10 × 10 × 10 = <b>1000</b><br></p>
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<p>So logarithms aren't just whole numbers like 2 or 3: we found a value at <b>2.5</b>,</p>
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<p>We can find more values (using cube roots, fourth-roots etc) like 2.75, or 1.9055, and so on.</p>
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<p>But we don't have to use square roots etc to find logarithms, because ...</p>
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<div class="center80">
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<p class="center larger">... <b>in practice</b> it is easier to use a calculator!</p>
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</div>
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<p> </p>
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<h2>Just Use A Calculator</h2>
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<table style="border: 0;">
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<tr>
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<td><img src="../money/images/calculator-log.gif" alt="log" width="114" height="91" /></td>
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<td> </td>
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<td><p>For example the "log" button will give the "base 10" logarithm.</p></td>
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</tr>
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</table>
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<div class="example">
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<h3>Example: Using the calculator, what is <span class="large">log<sub>10</sub>(300)</span> ?</h3>
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<p>Get your calculator, type in <b>300</b>, then press <b>log</b></p>
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<p>Answer: <b>2.477...</b></p>
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<p> </p>
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<p>That means that we need to use 10 in a multiplication 2.477... times to make 300:</p>
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<p class="center large">log<sub>10</sub>(300) = 2.477...</p>
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<p>Our earlier estimate of <b>2.5</b> wasn't too bad, was it?</p>
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<p>Note: using exponents it is: <b>10<sup>2.477...</sup> = 300</b></p>
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</div>
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<p> </p>
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<div class="example">
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<h3>Example: What is<span class="large"> log<sub>10</sub>(640)</span> ?</h3>
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<p>Get your calculator, type in 640, then press log</p>
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<p>Answer: <b>2.806...</b></p>
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<p>That means that we need to use 10 in a multiplication 2.806... times to make 640:</p>
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<p class="center large">log<sub>10</sub>(640) = 2.806...</p>
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<p>Have a look at the graph above, and see what value you get at x=640</p>
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<p>Note: using exponents it is: <b>10<sup>2.806...</sup> = 640</b></p>
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</div>
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<p> </p>
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<p>So there you have it ... logarithms (that tell us how many times to use a number in a multiplication) can have decimal values.</p>
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<p> </p>
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<div class="questions">
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<script>getQ(2388, 2389, 8938, 8939, 8935, 8936, 8937, 8940, 8941, 8942);</script>
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<a href="index.html">Algebra Index</a>
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