exponential growth

Suppose a colony of bacteria doubles every hour. After one hour you have twice as many, after two hours four times as many, after three hours eight times. The increase gets bigger and bigger because each new amount is a percentage of an already larger amount. This is exponential growth: growth by a constant percentage (or factor) per unit of time, not a constant amount.

We model it with f(t) = a · b^t, where a is the starting amount, b is the growth factor (with b > 1), and t counts the units of time. A 5% increase per year, for example, means b = 1.05, so each year the quantity is multiplied by 1.05. The slow-looking early stages can hide an explosive future — this is why exponential growth so often surprises people.

It helps to know the difference from linear growth, which adds the same amount each step (a straight line). Exponential growth multiplies, so it always overtakes any linear pattern eventually. Real-world quantities cannot grow exponentially forever — food, space, or money run out — so such models hold only while resources are effectively unlimited.