From 11:00PM PDT on Friday, July 1 until 5:00AM PDT on Saturday, July 2, the Shmoop engineering elves will be making tweaks and improvements to the site. That means Shmoop will be unavailable for use during that time. Thanks for your patience!

# Logarithms and Exponential Functions

Limits of Exponential Functions

# Limits of Exponential Functions

Everyone has their limit; logs and exponents are no different.

Let's look at the exponential function f (x) = 4x. No matter what value of x you throw into it, you can never get f (x) to be negative or zero. (How optimistic of it.) Try a few:

42 = 16
43 = 64
44 = 256

40 = 1
4-2 = 1/16

That's reflected in the function's graph, too. It never dips below y = 0.

### Sample Problem

What's an exponential function that contains the following set of points? Hint: remember that anything to the power of 0 becomes 1.

{(2, 18), (0, 2), (3, 54)}

Because any base raised to the power of 0 equals 1, we can easily figure out the coefficient. The point (0, 2) means that y is 2 when x is 0, or f (0) = 2. Throw that bad boy into the mix and solve:

f (x) = Cbx
2 = Cb0
2 = C(1)
C = 2

Now that you know C, use it to find the base. We'll use the point (2, 18), which means f (2) = 18:

f (x) = 2bx
18 = 2b2
9 = b2

b = 3

Remember, we can't have a negative base. That's why we're only using the positive square root of 9, even though it's technically ±3.

So the exponential function we need is f (x) = 2(3)x. Let's double-check that it works with our third point, (3, 54).

f (x) = 2(3)x
f (3) = 2(3)3 = 2(27) = 54

Yep, we're in the clear.