Logarithms and Exponential Functions
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, in fact, that mathematicians named the exponential function after him. If we wanted to represent his doubling pay, we could use this function:
y = 2x
If we can think of linear functions as ones where you always add by the same amount, we can think of exponential functions as ones where you always multiple 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.)
When will the exponential function y = 2x and linear function y = 10x meet? Try graphing; you don't have to be exact.
It is important to note that in this sample, and quite often in life, exponential functions won't rise as quickly as linear functions will when x is small. It's not until right around x = 6 that these two functions meet up. They missed each other; one went to Paris, another to Berlin, but finally they are reunited. Oh, young love.
Depending on whom you ask, an exponential function either looks like this:
f(x) = Cbx
…or like this:
f(x) = bx
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.
b, the base, 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 has to be greater than 0. If it were negative, trying to plug in some exponents would give some non-real answers.