Doubling Time
Doubling time is the length of time required for an exponentially growing quantity to double in size.
Formula
T_d = \ln(2) / r
Definition
Doubling time is how long it takes for a growing quantity to become twice as large, the growth counterpart to half-life, and like half-life it is constant regardless of the starting amount, the signature of exponential behavior. For exponential growth $A(t) = A_0 \cdot e^{rt}$, the doubling time satisfies $A(T_d) = 2A_0$, giving $T_d = \ln(2)/r \approx 0.693/r$, and the Rule of 72 approximates this as $T_d \approx 72/r\%$ for percentage rates. Doubling time is scale-invariant, the time to go from $n$ to $2n$ equals the time from $1$ to $2$, a self-similarity that is a defining feature of exponential growth and connects to the fractal nature of geometric progressions; in ecology, the intrinsic rate of natural increase $r_m$ relates to doubling time the same way, $T_d = \ln(2)/r_m$.
Example
A city with $10\%$ annual population growth doubles roughly every $72/10 = 7.2$ years by the Rule of 72, doubling again $7.2$ years later. A bank account earning $5\%$ annual continuous interest has $T_d = \ln(2)/0.05 \approx 13.86$ years, close to the Rule of 72 estimate of $72/5 = 14.4$ years, so a $\$1{,}000$ deposit becomes $\$2{,}000$ in about $14$ years. Moore's Law historically described a doubling time of about $2$ years for transistor density, modeled as $A(t) = A_0 \cdot 2^{t/2}$, though physical limits have since slowed the trend.
Key Insight
The Rule of 72 is a mental math shortcut: for compound interest at $r\%$, dividing $72$ by $r$ estimates the years to double, so at $6\%$ money doubles in $12$ years and at $12\%$ in $6$ years. Doubling time is essentially half-life in reverse, and just as half-life is constant regardless of starting amount for decay, doubling time is constant regardless of starting amount for growth.