What is an exponential function?

An exponential function is a function of the form:


where equation is the base and equation is the exponent (see exponentiation). Usually when we talk of exponential functions, we mean the natural exponential function with the base equation. Euler’s number equation is a special number, just like equation.

Let us first have a look at what the function looks like when we plot it. The following graph shows equation:


One of the characteristics of exponential functions is the rapidly increasing growth as you can see in the graph. Further, any exponential function will always intersect the y-axis at 1. So equation for any exponential function regardless of its base (this is of course unless the function is a sum, for example equation in which case equation). This is useful to know when you want to plot an exponential function.

Changing the base of exponential functions

We have a rule to change the base of an exponential function. Let’s say we are given equation and we want to express the same function as equation with some other base equation. We can do this with the following rule:


So in this case, we can write equation and we now have the same function as equation in terms of base equation. One common case would be to change an exponential function to the natural exponential function by changing the base equation to the Euler number equation. In this case we use the natural logarithm equation: