## What is an exponential function?

An exponential function is a function of the form:

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

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

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 for any exponential function regardless of its base (this is of course unless the function is a sum, for example in which case ). 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 and we want to express the same function as with some other base . We can do this with the following rule:

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