How this instrument works
Doubling time answers one question: at a fixed growth rate, how long until a quantity is twice its starting size? The formula comes straight out of the differential equation for continuous growth, dN/dt = rN, whose solution is the exponential N(t) = N₀e^{rt}. Doubling means e^{rt} = 2, and taking the natural logarithm of both sides isolates the exponent: rt = ln 2, so t = ln 2 ⁄ r. The natural log shows up because it is, by definition, the inverse of e^x — the one operation built to undo exactly this kind of growth.
This exact formula describes continuous, moment-by-moment growth — bacteria dividing smoothly, a population model with no calendar, radioactive decay run in reverse. It is a close relative of, but not identical to, period-by-period compounding, where a rate applied once per year instead satisfies (1+r)ⁿ = 2, giving n = ln 2 ⁄ ln(1+r). Here is the counter-intuitive part: the folk-math 'Rule of 72' was never built to approximate this continuous formula — it approximates the discrete one, and the constant 72 was nudged up from ln 2 × 100 ≈ 69.3 specifically to correct for the gap between ln(1+r) and r at ordinary rates. At 7%, Rule of 72 actually lands nearer the true discrete answer than this page's own continuous formula does.
The formula breaks down at r = 0: dividing by zero growth gives no finite doubling time, which tracks — a quantity that isn't growing never doubles. A negative rate flips the reading rather than breaking the arithmetic; ln 2 ⁄ r comes out negative, and that figure is best read as a halving time for a shrinking quantity, not a doubling time run backwards.
- Enter the steady growth rate per period into the Growth rate (%) field — for instance 7, for 7% growth each period.
- The sheet converts that percentage to a decimal and applies t = ln(2) ⁄ r automatically; no separate unit conversion is needed.
- Read the outcome in Doubling time (periods) — the number of periods the quantity needs to reach exactly twice its starting value.
- Keep the period consistent throughout: a rate entered per year returns a doubling time in years, a rate per month returns months.
Worked example — a 7% growth rate
A process growing at a steady 7% per period — enter 7 into Growth rate (%) — has doubling time t = ln(2) ⁄ 0.07 = 0.6931471805599453 ⁄ 0.07 = 9.902102579427789 periods, the value the sheet reports in Doubling time (periods). Rounded for planning, that is just under ten periods for the quantity to reach twice its starting size.
Compare the shortcuts on the same 7%: Rule of 72 gives 72 ⁄ 7 ≈ 10.29 periods, and solving the discrete compounding equation (1.07)ⁿ = 2 exactly gives n = ln 2 ⁄ ln(1.07) ≈ 10.24 periods. This calculator's 9.90 is the continuous-growth answer — correct for its own model, noticeably smaller than either discrete figure, and a reminder to check which kind of growth a real problem describes before trusting a doubling-time number.
Questions
What does doubling time actually measure?
It is the number of periods a quantity needs, at a fixed growth rate r per period, to reach exactly twice its starting value: solve e^{rt} = 2 for t and you get t = ln(2) ⁄ r. At 7% per period that works out to about 9.90 periods, a touch below the roughly 10.3 periods a quick Rule-of-72 mental calculation (72 ⁄ 7) would suggest, since the two formulas model slightly different kinds of growth.
Why does the formula use ln(2) instead of a plain number?
Because doubling means multiplying by exactly 2, and the natural logarithm is the operation that turns a multiplication like ×2 back into an addition on the exponent. Solving e^{rt} = 2 requires taking the log of both sides, which is why ln(2) ≈ 0.693147 sits fixed in the numerator for every growth rate.
How does this differ from the Rule of 72?
Rule of 72 estimates period-by-period compounding, (1+r)ⁿ = 2, with 72 divided by the rate as a percent; this page solves the continuous-growth version, e^{rt} = 2, with ln(2) ⁄ r. At 7% the Rule of 72 gives about 10.29 periods, the exact discrete answer is about 10.24, and this continuous formula gives 9.90 — three related but genuinely different numbers.
What happens if the growth rate is zero or negative?
A zero rate divides by zero — correctly undefined, since a quantity that isn't growing never doubles. A negative rate returns a negative value, which is best read as a halving time for a shrinking quantity rather than a doubling time running into the past.
Does the growth rate need to be a whole percentage?
No — enter it as a percent in the Growth rate (%) field, decimals included, such as 7 or 7.5. The sheet divides by 100 internally before applying t = ln(2) ⁄ r, so the rate used inside the formula itself is a decimal like 0.07 or 0.075.
Can this be used for periods other than years?
Yes — the formula only needs a consistent rate per period; feed in a rate per year for a result in years, per month for months, or per generation for a population model. The Doubling time (periods) figure simply inherits whatever period the rate was measured in.