Logarithms and Exponential Functions

Exponential Growth and Decay

Exponential Growth and Decay

Sometimes we wonder how Expo got to be so big, so quickly. Some celebrity stars fall as quickly as they rise, like those one-hit wonders out there that everybody feels a little awkward mentioning. Even Expo might be here today, gone tomorrow. What makes all the difference between whether Expo grows or shrinks is his little ol' base.

Remember that exponential functions look a little like this:

f (x) = Cbx

The base b changes how the graph looks, depending on its value.

Why is that? Remember that exponential functions multiply their base by itself a number of times. This "number of times" is equal to the exponent. If we compare b > 1 and 0 < b < 1, here's what we get:

When b is bigger than 1, our y-values get larger and larger. When b is smaller than 1 (but greater than 0), our y-values shrink. That's why we say exponential functions where b > 1 represent exponential growth. On the other hand, exponential functions where 0 < b < 1 represent exponential decay.

Sample Problem

Does the function f (x) = 5(0.9)x represent exponential growth or decay?

All we've gotta do is look at the base. In this function, 5 is the coefficient, x is the exponent, and 0.9 is the base. Since 0.9 is between 0 and 1, we've got some exponential decay on our hands. Don't worry; it's way less gross than it sounds.

It's starting to seem like the base is the source of all of Expo's power, kind of like a horcrux. If that's the case, then he's an evil, immortal wizard who will stop at nothing to become the most powerful being in the universe. Could it be that Expo is really He-Who-Must-Not-Be-Named? If so, you've got to be prepared to face him.