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Allowance day: time to go buy that new video game / pair of jeans / crazy gadget. Wait, what's that, Mom? You want me to save *all* of my allowance now? Lame.

Every week you put away $15, watching the number tick up at the same rate: $15, $30, $45. This change can be easily represented by the linear function *f *(*x*) = 15*x*, assuming *x* represents weeks.

A linear function's got a constant rate of change. In other words, you *add* the same amount for every increase in *x*. Here you're adding $15 per week to your account.

We've seen these linear guys before. But what happens when we start *multiplying* by the same amount for every increase in *x*? Our new exponential buddies will have something to say about that. Read on, dear Shmooper.

### Linear and Exponential Growth

Expo doesn't deal with mere linear functions, though. He's a rising star. Every new movie seems to earn him more money; those producers just won't stop increasing his pay. In fact, the amount he earns seems to*double*every week. He's so popular that mathematicians named the**exponential function**after him. If we wanted to represent his doubling pay, we could use this function:*f*(*x*)= 2^{x}If we can think of linear functions as ones where you always add the same amount, we can think of exponential functions as ones where you always multiply by the same amount. Don't get caught dead calling it a "multiplicative function," though. That's something else completely. (For a fun challenge, try saying multiplicative three times fast.)

Depending on whom you ask, an exponential function either looks like this:

*f*(*x*)*= Cb*^{x}…or like this:

*f*(*x*) =*b*^{x}Essentially, it just comes down to whether Expo's got his coefficient sunglasses on or not. For the sake of generality, we'll keep the coefficient here. Too bad, though, 'cause Expo's eyes are so hypnotizing.

The base,

*b*, is very important to Expo's biz; he's got a few rules for how*b*'s gotta be.First, the base cannot be 1. If it were, we'd just have a function that always stayed the same no matter how large of an exponent you tried to stuff in there. That's not very awesome or very exponential-y.

Second, the base needs to be greater than 0. If it were negative, trying to plug in some exponents would give some non-real answers.

### 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*) =*Cb*^{x}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)represent exponential growth or decay?^{x}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.

### Solving Exponential Equations

It's time to get properly equipped so you can face off with Expo. Thankfully, exponential equations have a few weak spots that you can take advantage of in your epic battles to come.

Like we mentioned before, the base is the source of the exponential's power, so we'll focus on that. Here's the first lesson: Strike at the base.

For

*b*> 0, if*b*=^{x}*b*, then^{y}*x*=*y.*Yep, we can drop the base from both sides if they match. The reason this handy trick works, by the way, is because we can take the logarithm of both sides, which cancels out the bases.

*b*=^{x}*b*^{y}

log(_{b}*b*) = log^{x}(_{b}*b*)^{y}*x*=*y*For an equation like 4

^{y}= 4^{(x + 5)}, you could simply throw the 4s out the window to get*y*=*x*+ 5. (Make sure they don't hit anything on the way down. The neighbors won't be happy.)If the bases aren't the same, but share a common factor, you can just change the bases so they match. Just don't forget about your exponent properties. For example:

27

^{(4x + 1)}= 9^{2x}Both 27 and 9 have 3 as a factor, so let's change them both to powers of 3.

(3

^{3})^{(4x + 1)}= (3^{2})^{2x}3

^{(12x + 3)}= 3^{4x}Now that we've got matching bases on both sides, we can drop 'em and just keep the exponents.

12

*x*+ 3 = 4*x*8

*x*= -3*x*= -^{3}/_{8}### Sample Problem

Give it a shot: try solving for

*x*with this equation, where the bases don't equal each other:2

^{7x}= 8^{(2x + 7)}Start by turning that 8 into 2

^{3}.2

^{7x}= 2^{3(2x + 7)}2

^{7x}= 2^{(6x + 21)}7

*x*= 6*x*+ 21*x*= 21### Limits of Exponential Functions

Everyone has their limit; logs and exponents are no different.Let's look at the exponential function

*f*(*x*) = 4^{x}. 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:4

^{2}= 16

4^{3}= 64

4^{4}= 256

4^{0}= 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*) =*Cb*^{x}

2 =*Cb*^{0}

2 =*C*(1)*C*= 2Now that you know

*C*, use it to find the base. We'll use the point (2, 18), which means*f*(2) = 18:*f*(*x*) = 2*b*^{x}

18 = 2*b*^{2}

9 =*b*^{2}*b*= 3Remember, 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). Let's double-check that it works with our third point, (3, 54).^{x}*f*(*x*) = 2(3)^{x}*f*(3) = 2(3)^{3}= 2(27) = 54Yep, we're in the clear.

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