Proportional and Relative Changes

There is an excellent description of the difference between logarithmic and linear scales in the book “Mathematical Analysis for Economists” by R Allen which I describe here.

Consider the following two sequences of numbers:

100,  150,  200,  250,    300,     \ldots
100,  150,  225,  337.5, 506.25, \ldots

the first sequence shows a regular increase of 50 units and the second a regular increase of 50 per cent from one number to the next. On a linear scale, the points representing the first sequence appear as equal distances from each other and those representing the second sequence at increasing distances. If we now take the logarithm to the base 10 of each number we will generate the following two new sequences:

2,  2.176,  2.301,  2.398,  2.477,  \ldots
2,  2.176,  2.352,  2.528,  2.704,  \ldots

What is striking here is that it is the second sequence that now shows points at equal distances (1.176 apart). It would seem, therefore, that equal distances between points on a linear scale indicate equal absolute changes in the variable and equal distances between points on a logarithmic scale indicate equal proportional changes in the variable. Before taking the logarithm, the second sequence increased by 50% each time, in log form however, it increased by a constant absolute amount of 1.176.

We can show this property quite easily as follows. If y_1, y_2, y_3 and y_4 are values shown by points at equal distances on a linear scale, then it must be true that

    \[y_3 - y_2 = y_2 - y_1\]

The same property on a logarithmic scale implies that

    \[\log y_3 - \log y_2 = \log y_2 - \log y_1.\]

Using the rule that the logarithm of a quotient is the difference of the separate logarithms:

    \[\log \frac{y_3}{y_2} = \log \frac{y_2}{y_1}\]

and taking anti-logarithms on both sides yields:

    \[\frac{y_3}{y_2} = \frac{y_2}{y_1}.\]

that is, in the linear scale,we have equal proportional changes in the variable. In practical terms, if we plot an exponential growth curve on a semi-logarithmic scale, we will observe a linear relationship because exponential growth means a fixed percentage change in the variable. This example why the following relationship is true:

    \[\frac{d\ln y}{dx} = \frac{1}{u} \frac{du}{dx}.\]

that is the change of a variable, y, in logarithmic space is equal to a proportional change of the same variable in linear space.

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