7: Log transformation straightens out exponential growth

transformer robot as truck


What we did on the graph is called log transformation, which 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 tick mark on the axis increases the number by the same amount — for example, the tick marks might stand for 0, 5, 10, 15, etc.

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

transformer robot transformed

When you transform data, you are putting it on a log scale. The tick marks on a log scale represent numbers that differ by the same FACTOR. And in fact our bacterial populations  differ by the same factor  after each generation — 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:

When log-transformed:

ANY population that grows or shrinks by a constant multiplicative factor will appear as a straight line (pointing up or down)

ANY population that is constant will appear as a flat line.