Solve an Exponential Equation for the Exponent

An unknown stuck in an exponent cannot be reached by dividing or subtracting; you have to take logarithms of both sides. Divide off the coefficient first, then apply the power rule: a·bˣ = y becomes bˣ = y/a, and x = ln(y/a) / ln b. Worked example: 5 · 2ˣ = 320 gives 2ˣ = 64, so x = ln 64 / ln 2 = 6. The natural log is only a convenience — base 10 gives the identical answer, because the two conversion factors cancel.

This one manoeuvre answers a whole family of questions that otherwise need guesswork: how many years until an investment reaches a target, how long until a drug concentration falls below a threshold, how many half-lives a sample has been through. Napier built his tables in 1614 for precisely this reason, and the Rule of 72 that bankers still quote is nothing more than ln 2 ≈ 0.693 rounded up and dressed for mental arithmetic. Traps to watch: divide by the coefficient before logging, since ln(5 · 2ˣ) is ln 5 + x ln 2, not 5x ln 2; and a base of exactly 1 makes the equation unsolvable, because 1ˣ never budges from 1.