Exponential Growth Calculator: Growth and Decay, Rate and Doubling Time
Find what a quantity grows or shrinks to after a number of periods, or the rate that took it from one value to another, with the doubling time or half-life and a period-by-period table.
Exponential growth multiplies a quantity by the same factor every period. Growing by a percent each period, the value after t periods is start x (1 + r/100)^t, so 1,000 growing 5% a period for 10 periods reaches 1,628.89. For continuous growth the formula is start x e^(rt/100). A negative rate gives decay, and the calculator also reports the doubling time or half-life.
The calculator assumes the rate stays exactly the same for every period, which real quantities rarely do over long spans. The result is what the model predicts, not a forecast and not professional advice, and over long spans a constant rate is the first assumption to question.
Ending value
1,628.89
- Change
- 628.89
- Total change
- 62.89%
- Growth factor per period
- 1.05
- Doubling time
- 14.21 periods
Formula: start x (1 + rate/100)^time
| Period | Value |
|---|---|
| 0 | 1,000 |
| 1 | 1,050 |
| 2 | 1,102.5 |
| 3 | 1,157.63 |
| 4 | 1,215.51 |
| 5 | 1,276.28 |
| 6 | 1,340.1 |
| 7 | 1,407.1 |
| 8 | 1,477.46 |
| 9 | 1,551.33 |
| 10 | 1,628.89 |
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About the Exponential Growth Calculator
A quantity grows exponentially when each period adds a fixed share of what is already there, rather than a fixed amount. OpenStax College Algebra 2e (section 6.1) writes this as f(x) = a b^x, with a the start and b the growth factor per period. If the quantity rises by r percent each period, the factor is 1 + r/100, which matches the same section's compounding formula, A(t) = P(1 + r/n)^(nt), with n = 1. At 5% the factor is 1.05, and 1,000 becomes 1,628.89 after 10 periods, a change of 628.89. Continuous growth uses the same section's formula A(t) = a e^(rt), with your rate divided by 100. Enter a negative rate for decay: a shrinking population or a radioactive sample. Doubling time follows section 6.7: set the end to twice the start, take the natural log, and t = ln 2 / k, where k is the continuous rate. At 5% a period, k is ln 1.05, so the doubling time is ln 2 / ln 1.05, or 14.21 periods. For decay, the same step gives the half-life.
Frequently asked questions
What is the difference between percent growth and continuous growth?+
Percent growth adds the rate once per period, so 10% a period turns 1,000 into 1,100 after one period. Continuous growth compounds at every instant, so a continuous 10% gives 1,000 x e^0.1, which is 1,105.17 after one period. That is the worked example in OpenStax section 6.1. Use continuous when your rate is quoted as a continuous rate, as in the radon-222 example in the next answer.
How do I calculate exponential decay?+
Enter the rate as a negative number. With percent decay at -50% a period, 1,000 drops to 500 after one period and 250 after two, so the half-life is exactly one period. For a continuous rate, OpenStax works radon-222 at a continuous 17.3% a day: 100 mg falls to about 59.51 mg after 3 days.
How do I find the growth rate between two values?+
Choose growth rate as the thing to solve for, then enter the starting value, the ending value and the number of periods. For percent growth the rate is 100 x ((end / start)^(1/t) - 1); for continuous growth it is 100 x ln(end / start) / t. Both values must be above zero.
What units does the time use?+
Whatever unit your rate uses. If the rate is per year, the time is in years; if it is per hour, the time is in hours. The doubling time and half-life come out in the same periods. Fractional times such as 2.5 are fine, and the table adds that time as its last row when it is 60 periods or less.
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