AP Calculus AB and BC glossary

Doubling time

Doubling time is how long an exponentially growing quantity takes to reach twice its current size. It equals the natural log of 2 divided by the growth constant. The starting amount cancels out of the equation, so the same interval doubles the quantity again from any moment onward.

Solve $y_{0}e^{kt} = 2y_{0}$ . The starting amount cancels, leaving $e^{kt} = 2$ , so $kt = \ln 2$ . Nothing about the answer depends on how large the quantity was when you started the clock.

t_{d} = \frac{\ln 2}{k}, \qquad k > 0

Since $\ln 2 \approx 0.693$ , a growth constant of $k = 0.05$ per year gives $t_{d} = \frac{0.693}{0.05} \approx 13.9$ years. That is the arithmetic behind the rule of 70 that finance classes quote, and it is a fast sanity check on any exponential answer you produce.

The mistake

Scaling it linearly. Tripling takes $\frac{\ln 3}{k}$ , which is about $1.585$ doubling times, not $1.5$ . Doubling time is also specific to pure exponential growth: a logistic model has no fixed doubling time, because growth slows as the population nears its carrying capacity.

Appears in: Unit 7: Differential Equations