Exponential Growth and Decay Calculator

Exponential change means the value is multiplied by the same factor in every equal stretch of time. Grow by 5% a year and each year's amount is 1.05 times the year before, so the growth itself keeps getting bigger. Decay by 5% a year and each year's amount is 0.95 times the last, so it falls quickly at first and then more and more slowly, never quite reaching zero.

The formula

final value x(t) = starting value x₀ × (1 + rate r)^time t
for decay, use a negative rate: x₀ × (1 − r)^t
starting value x₀ = x(t) ÷ (1 + r)^t
rate r = (x(t) ÷ x₀)^(1 ÷ t) − 1
time t = ln(x(t) ÷ x₀) ÷ ln(1 + r)
continuous form: x(t) = x₀ × e^(k × t), with k = ln(1 + r) and r = e^k − 1
doubling time = ln 2 ÷ ln(1 + r) = ln 2 ÷ k (ln 2 ≈ 0.693147)
half-life (decay) = ln 2 ÷ −ln(1 − r)
rule of 70: doubling time ≈ 70 ÷ rate in %
rate from two points: r = (x₂ ÷ x₁)^(1 ÷ (t₂ − t₁)) − 1

Worked example

Final value (the default): 1,000 growing 5% a year for 10 years. The growth factor is 1 + 0.05 = 1.05. Raise it to the time: 1.05¹⁰ = 1.6288946 (to 7 decimal places). Times the start: 1,000 × 1.6288946 = 1,628.8946, or 1,628.89, an increase of 628.89 or 62.89%.

Doubling time: ln 2 ÷ ln 1.05 = 0.6931472 ÷ 0.0487902 = 14.2067 years. The rule of 70 gives 70 ÷ 5 = 14 years, and the rule of 72 gives 72 ÷ 5 = 14.4 years.

Continuous rate: k = ln 1.05 = 0.0487902, so growing 5% a year in steps is the same as growing 4.879% a year continuously. Going the other way, a continuous 5% for 10 years gives 1,000 × e^0.5 = 1,000 × 1.648721 = 1,648.72, which is 19.83 more than the step version.

Starting value: how much would you need to start with to have 2,000 after 10 years at 5%? 2,000 ÷ 1.6288946 = 1,227.83.

Time: how long does 3 take to reach 24 when it doubles (+100%) every period? 24 ÷ 3 = 8, and ln 8 ÷ ln 2 = 3 periods, since 2 × 2 × 2 = 8.

Rate from two points: 200 at time 2 and 450 at time 6. 450 ÷ 200 = 2.25, over 6 − 2 = 4 periods. The 4th root of 2.25 is 1.224745, so the rate is 22.47% per period, and the value at time 0 was 200 ÷ 1.224745² = 200 ÷ 1.5 = 133.33.

Decay: 1,000 falling 5% a year for 10 years: 1,000 × 0.95¹⁰ = 1,000 × 0.5987369 = 598.74. The half-life is ln 2 ÷ −ln 0.95 = 0.6931472 ÷ 0.0512933 = 13.5134 years (rule of 70: 14).

Rule of 70 vs the exact doubling time

The rules work best for small rates: from 1% to 10% the rule of 70 is within about 0.35 periods of the exact answer. Above about 20% use the exact answer.

Notes

• The rate and the time can use different units. A month here is 1/12 of a year and a year is 365.25 days, so 2 years at a monthly rate is exactly 24 months.
• “Periods” is a plain step, such as a generation or a round. Use it for both the rate and the time.
• Fractional periods (like 2.5 years) follow the smooth curve: 1,000 × 1.05^2.5 = 1,129.73.
• The examples use made-up round numbers to show the maths. They are not data about any real town, disease, app or product.
• For money with regular deposits or monthly compounding, the compound interest calculator is a better fit. For radioactive decay, try the half-life calculator.