Solving Exponential Equations

An exponential equation is simply an equation where the unknown is up in the exponent, like $3^x = 20$: "3 to what power gives 20?" That's exactly the question a logarithm answers, which is why the last lesson came first. The formula up top just says it with symbols: $b$ (the base), $c$ (the number you want to reach), and $x = \frac{\ln c}{\ln b}$ (the power that does it). It's one idea used a few different ways. You'll use it for every "how long until...?" question about growth or decay.

Same base. Sometimes you can skip logs entirely. In $2^{x + 1} = 8$, notice $8 = 2^3$, so the equation says $2^{x + 1} = 2^3$. Two powers of 2 are equal only when the exponents are equal, so $x + 1 = 3$. This works when both sides are obviously powers of one number.

Logarithms. When the numbers don't cooperate, as in $3^x = 20$, take $\ln$ of both sides. The power rule $\log x^r = r\log x$ brings the $x$ down where you can reach it: $x\ln 3 = \ln 20$, so $x = \frac{\ln 20}{\ln 3}$.

Quadratic in disguise. $e^{2x} - 3e^x + 2 = 0$ looks nasty, but $e^{2x} = (e^x)^2$, so with $u = e^x$ it's just $u^2 - 3u + 2 = 0$. Solve for $u$, then go back to $x$. Throw out any solution with $e^x \le 0$, since $e^x$ is always positive.

A common mistake. In $5e^{2x} = 40$, it's tempting to take $\ln$ right away. But $\ln(5e^{2x})$ is $\ln 5 + 2x$, not $2x\ln 5$ or anything tidy. Isolate the exponential first, dividing by 5 to get $e^{2x} = 8$, and then take logs.

"How long until..." questions are exponential equations: doubling time of an investment, time for a drug to drop below a threshold, the age of a fossil from its carbon-14.