Exponential Growth Calculator
Enter a starting value, a per-period growth rate, and a time span — the calculator compounds x(t) = x₀(1 + r)ᵗ and shows total change and doubling time.
Example: with Starting value (x₀) 1000 · Growth rate per period (%) 5 · Number of periods (t) 10 → Final value x(t): 1,628.89.
Computed by the calculator below using its default values. Change any input to see your own numbers.
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Check it outThe exponential growth formula
Exponential growth multiplies by the same factor every period: x(t) = x₀(1 + r)ᵗ, where r is the per-period rate as a decimal. At 5% per year the growth factor is 1.05, so $1,000 becomes 1,000 × 1.05¹⁰ = $1,628.89 after 10 years — a 62.9% gain, not the 50% you would get by adding 5% ten times. The same equation is the standard population growth model: a town of 20,000 growing 2% a year reaches 20,000 × 1.02²⁵ ≈ 32,812 people in 25 years.
Doubling time and the rule of 70
Any steady growth rate implies a fixed doubling time: t₂ = ln 2 ÷ ln(1 + r). At 5% per period that is 0.6931 ÷ 0.04879 ≈ 14.21 periods; the “rule of 70” shortcut (70 ÷ 5 = 14) lands close. This is why an exponential growth graph looks deceptively flat at first and then rockets upward in a J-shape — each doubling adds as much as all previous growth combined. Negative rates run the same math in reverse: at −10% per period a value halves every 6.58 periods.
How it’s calculated
x(t) = x₀ × (1 + r⁄100)ᵗ, with r the percentage growth rate per period and t the number of periods (fractional t allowed). Total change = x(t) − x₀, also shown as a percentage of the start. Doubling time = ln 2 ÷ ln(1 + r⁄100) periods for positive rates; for negative rates the halving time ln 0.5 ÷ ln(1 + r⁄100) is shown instead. The rate compounds — it is applied to each period’s new total, not the original value.
Results update as you type and are estimates, not professional advice — verify important decisions with a qualified professional.
Common mistakes
- Multiplying rate by time instead of compounding: 5% for 10 periods grows 62.9% (1.05¹⁰ = 1.6289), not 50%.
- Mismatched units — pairing a monthly growth rate with t in years; rate and periods must use the same time unit.
- Entering the growth factor as the rate: 5% growth means r = 5 here (factor 1.05), not r = 1.05.
Frequently asked questions
What is the exponential growth formula?
x(t) = x₀(1 + r)ᵗ: starting value times the growth factor raised to the number of periods. With x₀ = 500 and r = 100% per period, three periods give 500 × 2³ = 4,000.
How do I calculate population growth rate?
Use the same formula with r as the annual rate: population × (1 + r)ᵗ. Solve for r from two counts with r = (Pᵗ⁄P₀)^(1⁄t) − 1 — a town going 20,000 → 24,000 in 10 years grew about 1.84% per year.
How do I find doubling time?
Doubling time = ln 2 ÷ ln(1 + r). At 10% per period: 0.6931 ÷ 0.0953 ≈ 7.27 periods. The rule of 70 (70 ÷ rate in %) gives a quick estimate.
What does exponential growth look like on a graph?
A J-shaped curve: nearly flat early, then increasingly steep, because each period multiplies an ever-larger base. On a logarithmic y-axis it plots as a straight line.
How is this different from continuous growth (e^kt)?
Continuous compounding uses x = x₀e^(kt) with k = ln(1 + r). A 5% per-period rate corresponds to k ≈ 0.04879, so the two forms give identical results when converted properly.