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      <text>4.4.8 (Application - Logarithmic scales)</text>
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      <text><![CDATA[<h1>Logarithmic scales and large numbers</h1>
<p>We have seen that some of the most important properties of logarithms are the following \[ \ln(AB) = \ln(A) + \ln(B) \qquad \mbox{ and } \qquad \ln(a^b) = b \ln(a). \] This allows us to express a multiplication as addition of logs and it allows us to express a power as a multiplication. Note that these properties hold for logs of all bases, not just the natural logarithm. These properties allow us to substantially simplify many calculations because in both cases, they allow us to examine quickly growing numbers. For example the function \(y=2^x\) increases far faster than \(y=2x\), so, if we need to look at very large values of \(y=2^x\) it is easier to use logarithms to slow the growth of the function. This is called <strong>logarithmic scaling</strong>. This is a way of displaying numerical data over a very wide range of values in a compact way. Sometimes the largest data points can be hundreds or even thousands of times bigger than the smallest ones. Since we have seen that a wide variety of phenomena follow such exponential growth or decay, the use of log-scales becomes an important tool for plotting such relationships.</p>
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<p>Large numbers numbers are difficult for the human mind to deal with. For example we know that \(1000000\) is really big but so is \(1000000000\). What is the difference between these numbers? What does this look like? Whilst we can answer these questions mathematically, they are somewhat more difficult to understand intuitively. Since items that follow an exponential function can get very big very quickly, it is easy for us to loose our intuitive understanding of such phenomena. For example writing the above numbers as \(10^5\) and \(10^8\), it becomes far easier to compare them. The beauty of logarithms as we have seen is that they count multiplications as additions.</p>
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<p>Suppose that we have a relationship between two variables \(x\) and \(y\) governed by the exponential function \(y = ab^x\), where \(a,b &gt; 0\). Taking logs of both sides we obtain that \[ \begin{align*} \ln(y) &amp;= \ln(ab^x) \\ &amp;= \ln(a) + x\ln(b). \end{align*} \] If we were to plot this function we would see that it follows that usual exponential curve. However we could <em>scale</em> the \(y\)-axis so that \(Y = \ln(y)\). Then the \(Y\)-axis is called a <strong>logarithmic axis</strong> and the relationship between \(x\) and \(Y\) is given by 
\[ 
Y = (\ln(b))x +\ln(a). 
\] 
This is the equation of a straight line in the \((x,Y)\) plane and because equations of straight lines are simple, we can easily find the gradient which is \(\ln(b)\) and we can find the \(Y\)-intercept which is \(\ln(a)\). From these we can then easily find \(a\) and \(b\). See the plots below for more details.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="@@PLUGINFILE@@/Log%20scale.png" alt="image showing the difference between a normal and a log scaled graph" width="778" height="247" /></p>
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<file name="Log scale.png" path="/" encoding="base64">[encoded data, 101780 characters]</file>
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