Compound Interest, Doubling Time and Half-Life

A half-life says: every fixed stretch of time, the amount gets multiplied by one half. That means it never hits zero — half of something is always something. After one half-life you have half the original, after two you have a quarter, after three an eighth. The pattern is (1/2)ⁿ where n counts how many half-lives have passed. If the half-life is h years and t years have gone by, n = t/h, so the amount remaining is (initial amount) x (1/2)^t/h. Compound interest runs the same idea upward instead of down: an amount P growing at rate r per period becomes P(1 + r)ⁿ after n periods, because each period multiplies by the same growth factor (1 + r) instead of adding a flat fee.

Doubling time and half-life both come from solving for the exponent, which is exactly why logarithms show up here. Set P(1+r)^t = 2P, divide both sides by P, and the starting amount disappears — that's why doubling time only depends on the rate, never on how much you began with. The same cancellation happens with half-life: it depends only on the decay factor, not the sample size.

Compounding more often — monthly instead of yearly — splits the same annual rate into smaller pieces applied more times, nudging the total higher, since interest starts earning on interest sooner. Underneath all of this is a geometric sequence: a fixed starting value times a fixed ratio raised to a growing power, the same structure tested directly elsewhere, just dressed up in years and percentages instead of term numbers.