Exponential Growth

A bacterial culture looks flat for hours because early on, a constant percentage of a small number is still a small number. Ten percent growth on 100 cells is only 10 new cells. But that same 10 percent growth rate applied later to 10,000 cells adds 1,000 cells in one step. The rate never changed. The base it is acting on did. That is the whole mechanism behind exponential growth: a fixed per capita growth rate, applied repeatedly, multiplies the population rather than adding to it. Each new generation starts from a bigger number, so each identical percentage increase produces a bigger absolute jump. Plotted on a normal axis, this produces the classic J-shaped curve: slow, almost invisible growth at first, then a sudden, steep climb.

Because each time step multiplies the population by the same factor, plotting population against time on a log scale turns multiplication into addition. Multiplying by a constant factor every period means adding a constant amount to the log of the population every period, and adding a constant amount each step is exactly what a straight line does. So an exponential curve that looks explosive on a normal graph becomes a straight line when the vertical axis is logarithmic, which is a quick way to spot exponential growth in data.

Exponential growth describes what happens when resources are unlimited and nothing checks the population, which almost never holds for long in nature. It sets up the contrast with logistic growth, where growth slows as the population approaches the carrying capacity of its environment, and it explains why real populations eventually bend away from the J shape into an S shape.