Doubling Time
Functions & Advanced AlgebraDoubling 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. Like half-life in reverse, it is constant for exponential growth.
Example
A city with $10\%$ annual population growth: the Rule of 72 says doubling time $\approx 72/10 = 7.2$ years. After $7.2$ years, population doubles. Then doubles again $7.2$ years later.
Key Insight
Just like half-life is constant regardless of starting amount for decay, doubling time is constant regardless of starting amount for growth. This is the signature of exponential behavior.
Definition
For exponential growth $A(t) = A_0 \cdot e^{rt}$, the doubling time $T_d$ satisfies $A(T_d) = 2 A_0$, giving $T_d = \ln(2)/r \approx 0.693/r$. The Rule of 72 approximates: $T_d \approx 72/r\%$ for percentage rates.
Example
A bank account earning $5\%$ annual continuous interest: $T_d = \ln(2)/0.05 \approx 13.86$ years. Using Rule of 72: $72/5 = 14.4$ years (a close approximation). Starting with $\$1{,}000$, you'll have $\$2{,}000$ in about $14$ years.
Key Insight
The Rule of 72 is a mental math shortcut: for compound interest at $r\%$, divide $72$ by $r$ to estimate years to double. At $6\%$, money doubles in $12$ years; at $12\%$, in $6$ years.
Definition
Doubling time $T_d = \ln(2)/r$ is the solution to $e^{r T_d} = 2$. It is the multiplicative inverse of the growth rate in the log scale: $\ln(2)$ is the number of "growth units" needed. In ecology, the intrinsic rate of natural increase $r_m$ relates to doubling time by $T_d = \ln(2)/r_m$.
Example
Moore's Law historically described a doubling time of approximately $2$ years for transistor density. Modeled as $A(t) = A_0 \cdot 2^{t/2}$, where $t$ is in years. This predicts exponential growth but physical limits have slowed the trend.
Key Insight
Doubling time is scale-invariant: the time to go from $n$ to $2n$ equals the time from $1$ to $2$. This self-similarity is a defining feature of exponential growth and connects to the fractal nature of geometric progressions.