Exponential Growth & Doubling Time
N = N₀(1 + r)ᵗ or N₀eᵏᵗ: solve the final amount, the rate (CAGR) or the time to a target, with doubling time, rule of 72, half-life for decay and a table.
The final amount, rate or time you left blank, the growth factor and total change, exact doubling time against the rule-of-72 estimate, the equivalent continuous rate, and a period-by-period table.
Example: 1 000 growing 7 % a year is 1 967 after 10 years and doubles in 10.24 years (rule of 72 says 10.29); 1 000 → 2 000 in 10 years is a 7.18 % CAGR; −5 % a period halves every 13.5 periods.
A constant rate
compounds.
The two growth models, how each unknown is solved, and what the rule of 72 approximates.
The model
Growth by a fixed percentage per period is N = N₀(1 + r)ᵗ; growth at every instant is N = N₀·eᵏᵗ. The two describe the same curve with different rates: k = ln(1 + r), and a discrete rate of 7 % equals a continuous rate of 6.77 %. A negative rate is decay — the same formulas, with the amount falling. The page solves whichever of the final amount, the rate or the time you leave out: the rate as (N ÷ N₀)^(1/t) − 1 (the compound annual growth rate when the periods are years), the time as ln(N ÷ N₀) ÷ ln(1 + r).
Doubling time and the rule of 72
The exact doubling time is ln 2 ÷ ln(1 + r) periods, and the half-life for decay is the same with a minus sign. The rule of 72 estimates doubling time as 72 ÷ r %: at 7 % it gives 10.29 against the exact 10.24. It is a mental shortcut, most accurate between about 4 % and 12 %; for small continuous rates 69.3 ÷ r is the exact figure, and at high rates the rule drifts. Both are shown so you can see the gap.
What it assumes
A constant rate for the whole period — true of radioactive decay and compound interest, an approximation for populations, bacteria, page views or revenue, which change their rate. The table shows the curve period by period (or at evenly spaced steps for long runs). Money is not adjusted for inflation, fees or tax; the page is the arithmetic only. Nothing leaves the browser; the same four anonymous usage counts as the rest of the site apply.
SOURCES
- Discrete N = N₀(1 + r)ᵗ, continuous N = N₀eᵏᵗ; doubling time ln 2 ÷ ln(1 + r); rule of 72 = 72 ÷ r %; continuous rate k = ln(1 + r); rate from two values r = (N/N₀)^(1/t) − 1
Last reviewed 22 September 2026. How results are checked: How we verify.