# What is Exponential Growth?

Jul 22, 2015

$y$ is an exponential growth function of $x$ if $y = a \cdot {b}^{x}$ for some $a > 0$ and some $b > 1$. This is often written as $y = a \cdot {e}^{k \cdot x}$, where $e = 2.718281828459 \ldots$ and ${e}^{k} = b$ (so $k = \ln \left(b\right) > 0$).

#### Explanation:

This is the definition of what an exponential growth function is.

On the other hand, if $0 < b < 1$, then the function is called an exponential decay function (and $k = \ln \left(b\right) < 0$).

You should graph examples of functions like these on your calculator to see what their graphs look like.

The number $e = 2.718281828459 \ldots$ is a "special" number in mathematics (special like $\pi$ is special). When it's used as the base of an exponential function in calculus, the resulting calculations you can do with it are simpler than they would be if you used some other base.

The function $\ln \left(x\right) = {\log}_{e} \left(x\right)$ is called the "natural logarithm" and the same comments apply to it.