The situations we have looked at that are characterized by exponential change can be seen as functions. In each situation, there is a quantity—an independent variable—that determines another quantity—a dependent variable. They are functions because any value of the independent variable that makes sense corresponds to only one value of the dependent variable. Functions that describe exponential change are called exponential functions.

For example, suppose $t$ represents time in hours, and $p$ is a bacteria population $t$ hours after the bacteria population was measured. For each time $t$, there is only one value for the corresponding number of bacteria, so we can say that $p$ is a function of $t$ and we can write this as $p = f(t)$.

If there were 100,000 bacteria at the time it was initially measured and the population decreases so that $\frac{1}{5}$ of it remains after each passing hour, we can use function notation to model the bacteria population:

$\displaystyle f(t) = 100,000 \boldcdot \left(\frac{1}{5}\right)^t$

Notice the expression in the form of $a \boldcdot b^t$ (on the right side of the equation) is the same as in previous equations that we wrote to represent situations characterized by exponential change.