Exponential Functions & Equations

Growth and decay with a b^x, and solving a b^x = c with logs.

The idea

An exponential function models a quantity that each step multiplies by a fixed amount, as a linear function models one that each step adds a fixed amount to. Its form is $f(x) = a\,b^{x}, \qquad b > 0, \quad b \neq 1.$ Raising the input by $1$ multiplies the output by the base $b$, and setting $x = 0$ gives $f(0) = a$, the starting value. When $b > 1$ the values climb, called growth; when $0 < b < 1$ they shrink, called decay. Doubling every hour is base $2$, halving is base $\tfrac{1}{2}$, and ten percent yearly interest is base $1.1$. The base $1$ is excluded because multiplying by it changes nothing, and negative bases because a power such as $b^{1/2}$ would have no real value.

An exponential equation, $a\,b^{x} = c$, asks for the input that produces a given output. Divide by $a$ so the power stands alone. If $c/a$ is a recognisable power of $b$, match exponents, since $b^{x} = b^{y}$ if and only if $x = y$; otherwise apply the logarithm, which undoes the base-$b$ power: $x = \log_{b}(c/a)$.

Ways to work on it

• Walkthrough. The shape of b^x, growth vs. decay, and solving with matched exponents.
• Practice. Evaluate an exponential model or solve a clean power equation.
• Hardest. Solve a b^x = c exactly for x.

Not sure where to start? Take the ten-question placement test.