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Exponential Growth Calculator

Model anything that grows or shrinks by a fixed percentage: populations, bacteria, investments, depreciation. Enter the starting amount, rate and time to get the final value, or solve for the rate or the time, with discrete or continuous compounding.

Solve for

Final amountRateTime

Type

GrowthDecay

Compounding

Initial amount P₀

Final amount P

Rate r per period

Time t

Time unit

Enter your values to see the result.

How to use the Exponential Growth Calculator

Choose what to solve for: final amount, rate or time.
Pick growth or decay and how often the rate is applied (once per period or continuously).
Enter the initial amount, the rate per period and the time (or the final amount).
Choose the time unit so the table and labels read naturally.
Press Calculate to see the result, doubling time or half-life, table and chart.

Exponential growth and decay formulas

Model	Final amount	Rate	Time
Discrete growth	P₀(1 + r)^t	(P/P₀)^1/t − 1	ln(P/P₀) ÷ ln(1 + r)
Discrete decay	P₀(1 − r)^t	1 − (P/P₀)^1/t	ln(P/P₀) ÷ ln(1 − r)
n times per period	P₀(1 + r/n)^nt	n((P/P₀)^1/(nt) − 1)	ln(P/P₀) ÷ (n ln(1 + r/n))
Continuous	P₀e^rt	ln(P/P₀) ÷ t	ln(P/P₀) ÷ r

Doubling time is the time where P/P₀ = 2, and half-life is where P/P₀ = ½, so both come from the time formula with ln 2 in the numerator.

Doubling time at different growth rates

Rate per period	Discrete: ln 2 ÷ ln(1 + r)	Continuous: ln 2 ÷ r	Rule of 72
1%	69.66	69.31	72
2%	35.00	34.66	36
3%	23.45	23.10	24
5%	14.21	13.86	14.4
7%	10.24	9.90	10.3
10%	7.27	6.93	7.2
15%	4.96	4.62	4.8
25%	3.11	2.77	2.9

The rule of 72 is most accurate for rates around 6% to 10%. For continuous growth the rule of 69.3 (100 × ln 2) is exact.

Frequently asked questions

What is the exponential growth formula?

For growth applied once per period, P(t) = P₀ × (1 + r)^t. For continuous growth, P(t) = P₀ × e^(rt). P₀ is the starting amount, r the rate per period as a decimal and t the number of periods.

What is the exponential decay formula?

Use a negative rate: P(t) = P₀ × (1 − r)^t for discrete decay or P₀ × e^(−rt) for continuous decay. A car worth 30,000 losing 15% a year is worth 30,000 × 0.85⁸ ≈ 8,175 after 8 years.

What is the difference between discrete and continuous growth?

Discrete growth adds the percentage at fixed steps (once a year, once a month). Continuous growth compounds at every instant and is slightly faster for the same nominal rate: 5% for 10 years gives 1.6289× discrete but e^0.5 = 1.6487× continuous.

How do you calculate doubling time?

Doubling time = ln 2 ÷ ln(1 + r) for discrete growth, or ln 2 ÷ r for continuous growth. The rule of 72 is a quick estimate: at 6% per year, 72 ÷ 6 ≈ 12 years.

How do I find the growth rate from two values?

For discrete growth r = (P / P₀)^(1/t) − 1; for continuous growth r = ln(P / P₀) ÷ t. Growing from 200 to 350 in 6 years is (1.75)^(1/6) − 1 ≈ 9.78% per year.

How do I find how long it takes to reach a target?

t = ln(P / P₀) ÷ ln(1 + r) for discrete growth, or ln(P / P₀) ÷ r for continuous growth. Choose "Time" in Solve for and the calculator does it for you.

Is the result a prediction?

It shows what happens if the rate stays exactly constant. Real populations, prices and investments rarely grow at a fixed rate for long, so treat the result as a model, not a forecast.

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