An exponential function is a mathematical expression in which a variable represents the exponent of an expression.

What does an exponential function look like?

Here's a very simple exponential function:

That equation is read as "y equals 2 to the x power."

Exponent refresher:

Let's remember how exponents work. Suppose we have the equation below:

That equation tells us to multiply x by itself to get y. It's the equivalent of:

If we want to find y when x=3, we can pretty quickly find that y=3*3=9. But, this is actually what's known as a "power function". In fact, it's just a polynomial, and not an exponential function at all.

Take a closer look at x² . This means x squared or x to the second power. What does it mean? There are two parts to this exponential term:

1) An exponent, which is the number 2.
2) A base, which is the variable x.

With exponential functions, the variable will actually be the exponent, with a constant as the base.

Exponential Functions

Here's what exponential functions look like:

The equation is y equals 2 raised to the x power. This sort of equation represents what we call "exponential growth" or "exponential decay." Other examples of exponential functions include:

The general exponential function looks like this: , where the base b is any positive constant. The base b could be 1, but remember that 1 to any power is just 1, so it's a particularly boring exponential function!

Let's try some examples:

Example 1

Solve for x: 

This one is actually pretty simple, so let's just think it through:

The problem says we have to multiply x number of two's together to get four. Well, everyone knows that 2*2=4, so the answer is two:

Ok, great, that was an easy example. But you can see where this could get really, hard, right? Look at this:

Example 2

Solve for y when x=5.

That means we need to plug-in x=5 and see what we get:

Fortunately all we had to do with this problem was multiply 1.2 times itself a few times to get the answer.

What about a word problem example?

We can use a formula for exponential growth to model the population of a bacteria. Let's say the bacteria population is defined by  where B is the total population and t represents time in hours. While that may look complicated, it really tells us that the bacteria grows by 12 percent every hour. Every time another hour goes by, t goes up by 1, so we have to multiply the population times 1.12 again. The 100 simply sets the initial population at time t=0.

So, how much bacteria remains after 4 hours?

What do we know? We have the formula  and the fact that t=4.

Replace t with 4 hours in the formula above and simplify.