Exponential Growth Calculator
This exponential growth calculator finds where a value ends up when it grows by a fixed percentage each period. Enter a starting value, a rate, and a time, and you'll see the final amount, the growth factor, and how long it takes to double. Switch between discrete and continuous models, and if you enter a negative rate it handles decay too, showing the half-life instead. There's a live curve so you can watch the growth take off.
- Discrete or continuous
- Growth and decay
- Growth factor
- Doubling time
- Live growth curve
Last updated June 18, 2026 Method: P = P₀(1+r)ₜ or P₀eⁿₜ Reviewed by the Calcowa math team
Enter a starting value, rate, and time to see the result.
P(t) = 100 × (1 + 0.05)¹⁰ = 162.89
What is the exponential growth formula?
Exponential growth means a value rises by the same percentage each period, and it's captured by P(t) = P₀ × (1 + r)ₜ for discrete steps, or P(t) = P₀ × eⁿₜ for continuous growth. Starting at 100 with a 5% rate over 10 periods, the discrete result is about 162.89.
The reason it accelerates is that each period's growth is based on the new, larger value, not the original. That's the snowball effect: the curve looks almost flat at first, then it shoots upward, and that's what catches people off guard. A negative rate flips the whole thing into decay, where the value keeps shrinking toward zero but it'll never quite reach it.
How do you calculate exponential growth?
It's a short routine once you've picked your model, and you won't need more than a calculator key for the power. Here's how it goes:
- 1
Note the starting valueWrite down P₀, the amount you begin with.
- 2
Convert the rate to a decimalDivide the percentage by 100, so 5% becomes 0.05.
- 3
Pick the modelUse (1 + r) raised to the time for discrete, or e raised to r times t for continuous.
- 4
Raise to the power of timeApply the exponent, then multiply by the starting value.
- 5
Read doubling time if you need itDoubling time is ln(2) / r continuous, or ln(2) / ln(1 + r) discrete.
Discrete vs continuous growth
The two models answer slightly different questions. Discrete growth, (1 + r)ₜ, fits anything that updates in steps: yearly interest, a population counted once a season, a savings balance. Continuous growth, eⁿₜ, fits things that change every instant, like bacteria dividing nonstop or interest compounded continuously. For the same rate, continuous always edges out a little higher, since the gains start compounding immediately rather than waiting. At low rates the gap's tiny, but it widens as the rate grows. If you're working with money specifically, the Compound Interest Calculator adds contributions and compounding intervals, and the Exponent Calculator handles the raw powers behind both formulas.
Growth factors at a glance
These show the discrete growth factor, how many times the starting value you'll end with. Notice how a higher rate compounds dramatically, and how a negative rate shrinks the value instead.
| Rate and time | Growth factor | Good to know |
|---|---|---|
| 5% for 10 periods | 1.629 × | Steady growth nearly doubles the start |
| 7% for 10 periods | 1.967 × | A small rate bump compounds fast |
| 10% for 10 periods | 2.594 × | Roughly 2.6 times the original |
| -10% for 5 periods (decay) | 0.590 × | A negative rate shrinks the value |
| 100% for 5 periods | 32 × | Doubling each period is explosive |
Frequently asked questions
Is exponential growth the same as compound interest?
They're the same math wearing different clothes. Compound interest is exponential growth applied to money, where the rate is the interest rate and the periods are compounding intervals. This calculator covers the general case for populations, science, and any percentage growth, while the compound interest calculator adds money-specific features like regular contributions.
There are two common forms. The discrete (periodic) version is P(t) = P₀ × (1 + r)ₜ, used when growth happens in steps like once a year. The continuous version is P(t) = P₀ × eⁿₜ, used when growth happens smoothly at every instant. Here P₀ is the starting value, r is the rate as a decimal, and t is the time.
Pick your rate and time, then raise the growth base to the power of the time. For 5% growth over 10 periods starting at 100, the discrete result is 100 × 1.05¹⁰, which is about 162.89. The continuous result is 100 × e⁰·⁵, about 164.87. The two stay close at low rates and drift apart as the rate climbs.
Discrete growth jumps at set intervals, like interest paid once a year, so it uses (1 + r)ₜ. Continuous growth compounds at every moment, like a population reproducing nonstop, so it uses eⁿₜ. Continuous always gives a slightly bigger result for the same rate, because the growth is being reinvested instantly rather than waiting for the next period.
Doubling time is how long it takes a value to double at a given rate. For continuous growth it's ln(2) / r, and for discrete growth it's ln(2) / ln(1 + r). At 5% continuous, that's about 13.86 periods. There's a handy shortcut too, the rule of 72: divide 72 by the percentage rate for a quick estimate.
Yes. Enter a negative rate and the same formulas describe decay instead of growth. At -10% over 5 periods, a starting value of 100 falls to about 59.05. For decay, the tool shows the half-life, the time for the value to drop by half, in place of the doubling time.
It's everywhere: populations, bacteria in a dish, compound interest, viral spread, and radioactive decay (as negative growth). Anything that grows or shrinks by a fixed percentage each period follows this curve. That percentage-of-itself behavior is what makes the growth start slow and then race upward.
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