The exact answer, and the shortcut
Anything that grows by the same percentage each period doubles on a fixed schedule. The exact time comes from logarithms; the rule of 70 is the back-of-envelope version.
The two agree closely at ordinary rates and drift apart only when the rate gets large, where the approximation behind the rule of 70 starts to strain.
How rate and time trade off
| Rate per period | Exact doubling | Rule of 70 |
|---|---|---|
| 2% | 35.0 | 35 |
| 7% | 10.24 | 10 |
| 10% | 7.27 | 7 |
| 100% | 1.00 | 0.7 |
The bottom row shows where the shortcut breaks: at 100% growth the quantity truly doubles in exactly one period, but 70/100 = 0.7 is well off. The rule of 70 is for small, steady rates.
Why it feels surprising
Exponential growth is hard to feel because early progress looks slow. A quantity doubling every ten periods barely moves for the first few, then adds more in the last period than in all the ones before it combined. The doubling time is the honest way to describe that pace in a single number.
Frequently asked questions
What is the doubling time formula?
t = ln(2) / ln(1 + r), where r is the growth rate per period written as a decimal. At 7% per period that is 0.6931 / ln(1.07) = 0.6931 / 0.06766 = 10.24 periods. It is exact for steady compound growth, unlike the rule-of-70 shortcut.
What is the rule of 70?
A mental shortcut: divide 70 by the growth rate in percent to estimate the doubling time. At 7% that gives 70 / 7 = 10 periods, close to the exact 10.24. It works because ln(2) ≈ 0.693 and, for small rates, ln(1 + r) ≈ r, so the ratio is about 69.3 / (100r). Some people use 72 because it divides more cleanly.
Why does a small rate change matter so much?
Because doubling time is inversely related to the rate. Growth of 2% doubles in about 35 periods; 4% doubles in about 18, not 33. Halving the rate roughly doubles the time, so small differences in a growth rate compound into large differences in outcome over the long run.
Does this work for decay or shrinking?
The same idea gives a half-life: how long to fall to half at a steady decline rate, using ln(0.5) / ln(1 + r) with r negative. This calculator focuses on growth, but the mathematics is the mirror image.
Is real growth ever this steady?
Rarely. Populations, prices and investments grow in fits and starts, so a constant-rate model is an approximation. It is still useful for seeing the shape of exponential growth and for quick comparisons, as long as you treat the doubling time as a guide rather than a promise.