doubling time

When something grows exponentially — savings at compound interest, a viral video's views, a young population — the most intuitive way to describe its speed is not an abstract growth rate, but a vivid question: how long until it doubles? That span is the doubling time, and it is the growth twin of the decay world's half-life.

For exponential growth y(t) = y0 e^(kt) with k > 0, the doubling time T_d is when y reaches 2 y0. Setting e^(k T_d) = 2 and taking logarithms gives k · T_d = ln 2, so T_d = (ln 2)/k ≈ 0.693/k. Just like half-life, it does not depend on the starting amount: it always takes the same time to double, whether you go from 100 to 200 or from a million to two million. A handy mental shortcut for percentage growth is the 'rule of 70' (or 72): doubling time in years is roughly 70 divided by the percentage growth rate, because ln 2 ≈ 0.70.

Doubling time makes exponential growth tangible and is the standard currency of finance, demographics, and epidemiology — early in an epidemic, the case count's doubling time is the headline number. It also makes the alarming power of exponential growth concrete: a steady 7% annual growth doubles in just ten years, quadruples in twenty, and is over a hundredfold in a single lifetime. The same constancy that makes it easy to quote is what makes unchecked exponential growth so explosive.