Doubling asks a different kind of question: not how much, but how long. The unknown moves into the exponent, and the only tool that reaches it is a logarithm. Set $A / P = 2$ in the growth relation and solve: $t = \dfrac{\ln(A/P)}{\ln(1 + r)}$ — t equals log of A over P, over log of one plus r, where $A$ is the final amount, $P$ the starting amount, $r$ the rate per period as a decimal, and $t$ the number of periods. Any log base works, as long as both logs use the same one.

The mental version is the rule of 72: $n \approx \dfrac{72}{R}$, where $R$ is the rate written as a percent — 6, not 0.06 — and $n$ is the number of periods to double. At 6% it says 12 years; the honest solve says 11.9. It is an approximation with a good excuse: $\ln 2 = 0.693$, and 69.3 is a miserable number to divide by, while 72 divides cleanly by 2, 3, 4, 6, 8, 9 and 12. The rounding error runs the right way for the rates people actually meet.

A third phrasing skips the rate entirely: $N = N_0 \cdot 2^{t/T}$, where $N_0$ is the starting amount, $T$ is the doubling time, $t$ is the elapsed time in the same units, and $N$ is the amount at the end. The exponent $t/T$ is nothing but a count of doublings.