8: Do ALL curves get “straightened” by taking the log?

Try it yourself and find out…

So, although log transformation is powerful, it is not ALL-powerful. The only curves that actually get straightened are the ones generated by functions that include “some number to the power of the x variable”. Why is that? Remember that:

any population that GROWS OR SHRINKS BY A CONSTANT MULTIPLICATIVE FACTOR will appear as a straight line when log-transformed.

And that’s what doubling does — it multiplies the population by a constant factor (2, that is) over a period of time. But other mathematical functions, such as the ones above, involve some sort of change in which the factor of change is NOT constant. Let’s take the parabola, for example: y = x2

x 1 2 3 4 5 6
y 1 4 9 16 25 36
factor 4 2.25 1.78 1.56 1.44

The factor by which y is increasing starts at 4 (4 divided by 1), then drops to 2.25 (9 divided by 4), then 1.78 (16 divided by 9), and so on. Although we might think that a parabola grows “pretty fast”, it is in fact “slow” compared to exponential growth that starts with the same factor.

As a general guideline:

  • if growth factors are increasing,
    then growth is proceeding faster than exponential,
    and that part of the log-transformed graph will
    curve up
  • if growth factors are decreasing,
    then growth is proceeding slower than exponential,
    and that part of the log-transformed graph will
    have a flattening curve
  • Only constant growth factors (exponential growth)
    leads to a straight-line when the graph is log-transformed.