Why does log-transformation unbend the plot of exponential growth?

 

Log transformation is a very powerful tool in biology, even if it cannot transform 18-wheelers or pickups trucks into sentient beings.

What you do when you log-transform data is change the scale of the data. Normal everyday data is usually expressed on a graph with a linear scale. Linear means that each tickmark on the axis increases the number by the same amount -- for example, the tickmarks might stand for 0, 5, 10, 15, etc.

However, our bacterial population is not increasing by the same amount in each timestep. Starting with 1 cell, in the first timestep we add 1 more, in the second timestep we add 2 more, in the third timestep we add 4 more, and so on. The graph does not look like a straight line.

 

When you transform data, you are putting it on a log scale, which is also called a multiplicative scale. The tickmarks on a multiplicative scale represent numbers that differ by the same FACTOR, or multiplier. And in fact our bacterial populations also differ by the same factor at each timestep -- they double, or increase by a factor of 2.

So when you use a log scale, populations separated by the same factor will line up in a straight line. The slope of that line will depend on how big the multiplying factor is, which we'll talk about in a few screens. But the crucial point is:

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