Exponential Functions

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:

$$ y=2^x $$

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

So, remind me what an exponent is...

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

$$ y=x^2 $$

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

$$ y=x*x $$

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 $x^2$. This means x squared or x to the second power. There are two parts to this exponential term:

1) An exponent, which is the superscripted 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.

Here's what exponential functions look like:

$$ y=2^x $$

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:

$$ y=3^x $$ $$ f(x)=4.5^x $$ $$ y=2^{x+1} $$

The general exponential function looks like this: $ \large y=b^x$, 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: $4=2^x$

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:

$$ 4=2^x $$ $$ x=2 $$

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 $x=5$.

$$ y=1.2^x $$

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

$$ y=1.2^5 $$ $$ y=1.2*1.2*1.2*1.2*1.2 $$ $$ y=2.48832... $$

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 commonly use a formula for exponential growth to model the population of a bacteria. Let's say a bacteria population is defined by $B(t)=100*1.12^t$ 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 $B(t)=100*1.12^t$ and the fact that $t=4$.

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

$$ B(t)=100*1.12^t $$ $$ B(4)=100*1.12^4 $$ $$ B(4)=100*1.12*1.12*1.12*1.12 $$ $$ B(4)=157.35... $$

The number actually has a lot of digits after the decimal place. A real problem would usually specify where you should round your answer, but in this case, rounding to the nearest WHOLE number makes sense. Why? Because we're dealing with bacteria here. There can only be a whole number of bacteria, so the answer is best expressed as 157 after 4 hours of growth.