Inverse, Exponential, and Logarithmic Functions: Exponential Growth and Decay

Exponential growth and exponential decay are both of the form

Q = Q₀e^kt

where Q₀ is the initial quantity, t is the time elapsed, and k is the rate constant.

k plays two roles. First, it determines whether the function will represent growth or decay. If k is positive, then the function represents growth. If it is negative, then the function represents decay.

Figure %: Exponential Growth (k positive) and Exponential Decay (k negative)

The second role that k plays is in setting the rate of growth or decay. The larger k is, the faster the rate of change.

With exponential growth, the rate of increase goes up with time. This should be apparent from the derivative:

Q₀ke^kt

Likewise, with exponential decay, the rate of decrease lessens with time.

To be more precise, one unique property of exponential growth and decay is that the rate of growth or decay is proportional to the value of the function. In other words, it has the property that:

= ky

What stays constant over time with a rate of change such as this is the percent increase of the function per unit time. Thus, something that grows at a rate of 20% percent per year exhibits exponential growth. The percent increase remains constant with time, but the rate of increase grows as the quantity grows.

It is in fact the case that all functions for which

= ky

is true are necessarily of the form Y = Y₀e^kt.