# Do ALL curves get "unbent" by taking the log?

Try it yourself and find out...

The only curves above that get straightened are the ones generated by functions that include "some number to the power of the x variable". Why is that? Recall:

ANY population that grows or shrinks by a constant factor will appear as a straight line when log-transformed.

All of the other functions involve some sort of change in which the factor of change is NOT constant. Let's take the parabola, for example: y = x2

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

The factor by which y is increasing starts at 4, then drops to 2.25, then 1.78, 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 curve will have a relatively steep slope.
• if growth factors are decreasing, then growth is proceeding slower than exponential, and that part of the log-transformed curve will have a relatively gentle slope.
• Only constant growth factors (exponential growth) leads to a straight-line when the graph is log-transformed.