The Doubling Time Formula, Explained Clearly

Where ln(2) comes from

If a quantity grows at a constant rate, its size after time T is the starting amount multiplied by (1 plus r) raised to the power T. Setting that equal to twice the starting amount and solving for T gives T equals ln(2) divided by ln(1 plus r). The ln(2) appears simply because we want the value to double.

For continuous compounding the algebra is even cleaner. The growth model is e to the power (r times T), and setting that equal to 2 gives T equals ln(2) divided by r. See Continuous Growth Rate Formula for the full derivation.

Discrete vs continuous

The discrete formula assumes the rate is applied once per period, like a yearly interest rate that posts on January first. The continuous formula assumes the rate is applied at every instant. In the real world, monthly compounding sits between the two but is very close to continuous for small rates.

If you switch between the two on the calculator, you will see the answers drift further apart as the rate gets larger. At 1 percent the gap is tiny. At 50 percent the gap is significant.

Sanity checks

Two shortcuts give you instant feedback. The Rule of 70 says doubling time is roughly 70 divided by the rate as a percentage. The Rule of 72 swaps 70 for 72, which divides cleanly into common interest rates. If your calculator answer is more than a percent or two off the matching rule, double check the inputs.

When the formula breaks down

The formula assumes a steady rate. If your real rate changes every year, the formula gives an average doubling time, not a guaranteed one. Use it for orientation, not promises.

For more on the rules of thumb, see Rule of 70 vs Rule of 72. For a step-by-step walkthrough, see How to Calculate Doubling Time.