Doubling Time

Doubling time is the t solving 2 = e^(rt) under continuous growth — a fixed span for a fixed rate, independent of the starting amount.

A population grows at 6% continuously; find its doubling time. Set the ratio to 2: 2 = e^(0.06t). Take the natural log of both sides — ln is e's own inverse, so it peels the exponential in one move: ln(2) = 0.06t. Divide: t = ln(2)/0.06 ≈ 0.6931/0.06 ≈ 11.55 years. The audit runs forward: e^(0.06 · 11.55) ≈ e^0.693 ≈ 2. And the independence claim earns its keep: no starting population appeared anywhere — the P canceled when the ratio was set to 2, which is why one rate owns one doubling time for towns of any size.