Exponential Functions

An exponential function is simply something that gets multiplied by the same amount every step, like money earning interest or a population doubling. The name comes from where the variable lives: in the exponent. In $f(x) = b^x$ there are just two pieces, $b$ (the base, what you multiply by each step) and $x$ (how many steps you've taken). The $e$ is just one particular base, about $2.718$, that turns out to be the most natural choice; we'll see why below. That's really all there is to the formula.

Linear vs exponential. You already know linear functions, which add the same amount each step: $2, 4, 6, 8$. An exponential multiplies instead: $2, 4, 8, 16$. They start out close, but multiplying runs away fast. Doubling 30 times turns 1 into more than a billion, while adding 2 thirty times gets you to 61.

Properties of $b^x$. Each one comes from the exponent rules in the first unit.

• $f(0) = b^0 = 1$, so every basic exponential passes through $(0, 1)$: zero steps, no multiplying yet.
• The output is always positive. Multiplying a positive number by a positive $b$ never gets you to zero or below, so the x-axis is a horizontal asymptote the curve approaches but never touches.
• It's increasing when $b > 1$ and decreasing when $0 < b < 1$, because multiplying by something less than $1$ shrinks.

The number $e$ comes from compounding more and more often: $\left(1 + \frac1n\right)^n \to e \approx 2.71828$ as $n \to \infty$. Think of $1$ dollar at 100% interest, compounded $n$ times a year; however often you compound, you never get past about $2.72$. It's the base that makes calculus simplest, because $e^x$ is its own derivative.

Compound interest, viral spread, radioactive decay, cooling coffee and the growth of computing power all follow exponential curves.