1 Logarithms and scaling

In this Section we shall work entirely with logarithms to base 10.

We are already familiar with a particular property of logarithms:   $log{A}^k=klogA$ .

Now, choosing $A=10$ we see that:   $log10^k=klog10=k$ .

The effect of taking a logarithm is to replace a power: $10^k$ (which could be very large) by the value of the exponent $k$ . Thus a range of numbers extending from 1 to 1,000,000 say, can be transformed, by taking logarithms to base 10, into a range of numbers from 0 to 6. This approach is especially useful in the exercise of plotting one variable against another in which one of the variables has a wide range of values.

Example 10

Plot the following values $\left(x,y\right)$

Estimate the value of $y$ when $x=1.35$ .

Solution

If we attempt to plot these values on ordinary graph paper in which both vertical and horizontal scales are linear we find the large range in the $y$ -values presents a problem. The values near the lower end are bunched together and interpolating to find the value of $y$ when $x=1.35$ is difficult.

Figure 12

Example 11

To alleviate the scaling problem in Example 10 employ logarithms to scale down the $y$ -values, giving:

Plot these values and estimate the value of $y$ when $x=1.35$ .

Solution

Figure 13

This approach has spaced-out the vertical values allowing a much easier assessment for the value of $y$ at $x=1.35$ . From the graph we see that at $x=1.35$ the ‘ $logy$ ’ value is approximately 1.05. Taking $logy=1.05$ and inverting we get

$y=10^{1.05}=11.22$