Exponential Growth Calculator
Exponential growth or decay by x₀(1 + r)^t or x₀e^(kt): solve the final amount, starting amount, rate or time, with the doubling time or half-life.
Result
Final amount: 162.8895
Exact for the model chosen; real populations and prices rarely grow at one steady rate.
- Growth factor, x(t) ÷ x₀
- 1.628895
- Change, x(t) − x₀
- 62.8895
- Doubling time, in periods
- 14.2067
- The same growth in the other model
- 4.879%
https://calcgallery.com/algebra-numbers/exponential-growth-calculator/
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How to use the Exponential Growth Calculator
- Choose growth by period, x₀(1 + r)^t, or continuous growth, x₀e^(kt).
- Choose what to solve for, then fill in the other three. Use a negative rate for decay.
- Keep the rate and the time in the same periods, such as a yearly rate and a number of years.
The two models
by period x(t) = x₀ × (1 + r)^tcontinuous x(t) = x₀ × e^(k × t)Each can be turned around to find whatever is missing. The starting amount is x(t) ÷ (1 + r)^t. The rate is (x(t) ÷ x₀)^(1/t) − 1 by period, or ln(x(t) ÷ x₀) ÷ t continuously. The time is ln(x(t) ÷ x₀) ÷ ln(1 + r), or ln(x(t) ÷ x₀) ÷ k.
When there is no answer
Some combinations cannot happen, and the page says so rather than printing a number. A shrinking quantity never grows to a larger target, and a growing one never falls to a smaller one. A rate of −100% by period wipes the whole amount out in the first period, so nothing is left to grow or shrink after that.
A steady rate is an assumption
Real populations slow down as they run out of room, and prices and audiences change their rates from year to year. The formula is exact for a quantity that really does change by the same percentage every period; for anything else it is the constant-rate version of the story, useful for comparing and for rough projections.
With your numbers
- x(t) = x₀ × (1 + r)^t: 162.889463 = 100 × (1 + 5%)^10
Quick reference
Exact for the model chosen; real populations and prices rarely grow at one steady rate.
| Example | Result |
|---|---|
| 100 at 5% for 10 periods | Final amount: 162.8895 |
| 100 at 5% continuous, 10 periods | Final amount: 164.8721 |
| 1,000 decaying 10% for 5 periods | Final amount: 590.49 |
| Rate to double in 10 periods | Growth rate per period: 7.1773% |
| Time to triple at 3% | Time, in periods: 37.167 |
| Time to triple at 3% continuous | Time, in periods: 36.6204 |
| Start for 500 after 8 at 4% | Starting amount: 365.3451 |
Common questions about the Exponential Growth Calculator
- What is the formula for exponential growth?
- By period, x(t) = x₀(1 + r)^t: the amount is multiplied by the same factor, 1 + r, every period. Continuous growth is x(t) = x₀e^(kt), where the change is compounded at every instant. Starting from 100 at 5% for 10 periods, the first gives 162.89 and the second 164.87.
- What is the difference between r and k?
- r is the growth over one whole period, and k is the continuous rate that produces it. They are related by 1 + r = e^k, so a continuous 5% is the same as 5.127% by period, and 5% by period is the same as a continuous 4.879%. The page shows the equivalent rate in the other model.
- How long does it take to double?
- ln 2 ÷ ln(1 + r) periods by period, or ln 2 ÷ k continuously. At 5% a period that is about 14.2 periods. The rule of 70, 70 ÷ the rate in percent, is a mental shortcut. For continuous growth it is almost exact, since ln 2 ÷ k is 69.3 ÷ k in percent. By period it drifts off as the rate rises, because ln(1 + r) falls further below r: at 20% it gives 3.5 periods against the true 3.8.
- How do I model decay?
- Enter a negative rate. Losing 10% each period is r = −10%, and 1,000 falls to 590.49 after five periods. The page then gives the half-life, the time to fall to half, instead of a doubling time. If you already know a half-life, the half-life calculator starts from it.
- Is this the same as compound interest?
- The arithmetic is the same, but the compound interest calculator works in money, with compounding periods and deposits. This page is for any quantity that changes by a steady percentage: a population, a bacterial culture, a spreading audience, a depreciating value.