Exponential Growth & Decay Calculator

Anything that changes by the same percentage of its current size in each unit of time follows an exponential curve: a bacterial culture, an unchecked population, money earning compound interest, a radioactive sample or a drug clearing from the blood. Enter any three of the initial amount, the rate, the time and the final amount, and the calculator finds the fourth. It also reports the doubling time or half-life, the growth factor per time unit, the equivalent rate in the other model, a table of values and a graph.

Rates are entered as positive percentages with a growth or decay switch, so a 12% decay is entered as 12 with Decay selected. The time unit selector only labels the answer: the rate must be per the same unit as the time.

Continuous: y = a·e^(rt) Per period: y = a(1 + r)^t

a is the initial amount, y the amount after time t, and r the rate per time unit as a decimal (negative for decay). The continuous model assumes the change happens at every instant, which suits cell growth and radioactive decay. The per-period model applies the rate once per period, as with annual interest or a yearly census. The two describe the same curve when r(continuous) = ln(1 + r(per period)).

Doubling time = ln 2 ÷ k, half-life = ln 2 ÷ |k|, with k = r or k = ln(1 + r)

Rearranging gives the other unknowns: a = y ÷ e^(rt), r = ln(y ÷ a) ÷ t (or (y ÷ a)^(1/t) − 1), and t = ln(y ÷ a) ÷ k. For a per-period decay the half-life uses ln(1 − rate), not ln(1 + rate); a 10% loss per period halves the amount in 6.579 periods, not the 7.273 it takes a 10% gain to double.

Bacteria that double every 30 minutes

Doubling every half hour means r = ln 2 ÷ 0.5 = 138.6294% per hour. Starting from 1,000 cells, after 3 hours there are 64,000 cells, six doublings or 2⁶ = 64 times the start.

Carbon-14 after 10,000 years

With a half-life of 5,730 years the decay constant is ln 2 ÷ 5730 = 0.0121% per year. Of 100 g, 29.83 g remains after 10,000 years, about 29.8% of the original.

An investment at 8% a year

$10,000 compounded once a year for 20 years grows to $46,609.57. Compounded continuously at the same nominal 8% it would reach $49,530.32, because continuous growth at 8% equals 8.3287% per year.

Finding a city's growth rate

A city grew from 500,000 to 750,000 people in 15 years. The continuous rate is ln 1.5 ÷ 15 = 2.7031% a year, and the equivalent annual rate is 2.74%. At that pace it doubles every 25.64 years.

A drug eliminated at 12% per hour

If 12% of a 200 mg dose is cleared each hour, 71.93 mg is left after 8 hours, and the half-life is 5.422 hours.

The rule of 70 (doubling time ≈ 70 ÷ rate in percent) matches the continuous column closely and drifts for large per-period rates.

Real growth slows when food, space or customers run out, so populations and epidemics follow an S-shaped logistic curve rather than an exponential one over long periods; treat long projections as what would happen if the rate stayed fixed. Decay never reaches exactly zero in the formula, while a real sample eventually runs out of atoms. Finance products add fees and different compounding schedules, so use the compound interest calculator for money, and the half-life calculator for isotope work with known half-lives.