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Exponential Growth Calculator and Decay

Doubling Time, Half-Life and Continuous Compounding for Steady Percentage Growth

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How the Exponential Growth and Decay Calculator works

This exponential growth calculator applies a fixed percentage change, period after period, to a starting amount. Pick Growth or Decay, and it returns the final amount, the growth or decay factor, the doubling time or half-life, and what continuous growth at the same rate would give.

Exponential change means the amount changes by the same percentage each period, not the same number. A population growing 5% a year adds more people each year because 5% of a bigger number is bigger. Decay works the same way in reverse: a drug cleared at 12% an hour loses less each hour.

How the exponential growth calculator works

The formula is A = P × (1 + r)^t for growth and A = P × (1 − r)^t for decay, where P is the starting amount, r the rate as a decimal and t the number of periods. The bracket is the growth factor. Doubling time is ln 2 ÷ ln(1 + r), and half-life is ln 2 ÷ −ln(1 − r).

First example. 1,000 growing at 5% for 10 periods: 1,000 × 1.05^10 = 1,000 × 1.6289 ≈ 1,628.89. It doubles every 14.21 periods.

Second example. 200 mg decaying at 12% per hour for 6 hours: 200 × 0.88^6 ≈ 92.88 mg left. The half-life is about 5.42 hours.

How many years does it take a town of 20,000 to reach 40,000 at 3% a year? That is one doubling, so ln 2 ÷ ln 1.03 ≈ 23.45 years.

Doubling time, half-life and 10-period change of 100 at rates from 1% to 25%

Rate per period Doubling time (growth) Half-life (decay) 100 after 10 periods of growth 100 after 10 periods of decay
1% 69.66 68.97 110.46 90.44
2% 35 34.31 121.9 81.71
3% 23.45 22.76 134.39 73.74
4% 17.67 16.98 148.02 66.48
5% 14.21 13.51 162.89 59.87
6% 11.9 11.2 179.08 53.86
7% 10.24 9.55 196.72 48.4
8% 9.01 8.31 215.89 43.44
9% 8.04 7.35 236.74 38.94
10% 7.27 6.58 259.37 34.87
11% 6.64 5.95 283.94 31.18
12% 6.12 5.42 310.58 27.85
13% 5.67 4.98 339.46 24.84
14% 5.29 4.6 370.72 22.13
15% 4.96 4.27 404.56 19.69
16% 4.67 3.98 441.14 17.49
17% 4.41 3.72 480.68 15.52
18% 4.19 3.49 523.38 13.74
19% 3.98 3.29 569.47 12.16
20% 3.8 3.11 619.17 10.74
22% 3.49 2.79 730.46 8.34
25% 3.11 2.41 931.32 5.63

Tips

FAQ

What is the formula for exponential growth?

A = P × (1 + r)^t. P is the starting value, r is the growth rate per period as a decimal, and t is the number of periods. At 8% per year for 5 years, 500 becomes 500 × 1.08^5 ≈ 734.66. The calculator shows each step of that sum in the working line.

How do you calculate half-life from a decay rate?

Divide ln 2 by the negative natural log of the decay factor. A 10% loss per period gives a factor of 0.9, so the half-life is 0.6931 ÷ 0.1054 ≈ 6.58 periods. After that many periods, half of whatever you started with remains, no matter how large it was.

What is the difference between exponential and continuous growth?

Ordinary exponential growth applies the rate once per period. Continuous growth compounds every instant and uses A = P × e^(rt). At 5% for 10 periods, 1,000 becomes 1,628.89 by steps and 1,648.72 continuously. The gap widens as the rate or the number of periods rises.

Is compound interest exponential growth?

Yes. Compound interest is exponential growth where the rate is the interest rate and each period is one compounding interval. The compound interest calculator adds regular deposits and compounding frequency; this page is the plain version for populations, bacteria, decay and prices.

See also: all Algebra and Geometry tools, Arithmetic, Number Theory, Statistics, Trigonometry and Calculus.

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