Free Exponential Growth Calculator

Last updated October 13, 2026 · Report an error

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Exponential growth describes any quantity that increases by a fixed percentage of its current size during each equal time period, rather than by a fixed absolute amount — which is exactly why exponential growth eventually overtakes any linear growth process, no matter how large the linear process's starting head start. This behavior appears throughout nature and society: bacterial and cell cultures dividing at a constant rate, populations growing at a steady percentage per year, the early stage of a viral outbreak's spread before any saturation sets in, and the value of a compounding investment. There are two closely related mathematical forms of this idea: "compound" (or discrete) growth, N(t) = N₀(1+r)ᵗ, which applies when growth happens in distinct steps (like once per year or once per generation), and "continuous" growth, N(t) = N₀·e^(rt), which applies when growth happens smoothly and constantly at every instant rather than in discrete jumps — the mathematical limit of compounding infinitely often. Both share the same underlying idea of a constant percentage growth rate, just modeled slightly differently depending on whether the process ticks in steps or flows continuously.

How it works

Enter the initial value N₀, the growth rate r (as a percentage per time period), and the elapsed time t, then choose compound or continuous growth. For compound growth, the calculator applies N(t) = N₀(1+r)ᵗ, raising (1 plus the rate) to the power of the number of elapsed periods. For continuous growth, it instead applies N(t) = N₀·e^(rt), using Euler's number e to model growth compounding at every infinitesimal instant rather than at fixed intervals. The calculator also reports the doubling time — how long it takes the quantity to reach exactly twice its starting value — found by solving each formula for the time at which N(t) = 2N₀: t = ln(2)/ln(1+r) for compound growth, or t = ln(2)/r for continuous growth.

Enter initial value (N₀).
Enter growth rate (%, per time period).
Enter time (t).
Enter growth type.
Click Calculate to see your results.

Examples

N0=100, r=5%, t=10, compound

100 × (1.05)^10 ≈ 162.89. The doubling time is ln(2)/ln(1.05) ≈ 14.21 periods.

N0=100, r=5%, t=10, continuous

100 × e^(0.05×10) = 100 × e^0.5 ≈ 164.87 — slightly higher than the compound result, since continuous compounding grows marginally faster than the same nominal rate compounded once per period.

N0=500, r=3%, t=20, compound

500 × (1.03)^20 ≈ 500 × 1.8061 ≈ 903.06.

Common mistakes to avoid

Mixing time units — entering an annual growth rate but a time period measured in months (or vice versa) gives a wildly incorrect result.
Entering the growth rate as a whole number percentage (like 5) instead of its decimal form (0.05) in a formula or spreadsheet expecting the decimal.
Assuming compound and continuous growth give identical results for the same nominal rate — they're close for small rates but diverge more noticeably as the rate or time period grows larger.
Extrapolating exponential growth far into the future without considering that real-world processes eventually hit resource or capacity limits that pure exponential models don't account for.

Frequently asked questions

The math is identical, but this tool is deliberately framed for any growing quantity — populations, cell cultures, viral spread, radioactive decay in reverse — rather than being labeled specifically for money, even though the same N₀(1+r)ᵗ formula applies equally well to a bank balance.

Compound growth assumes the quantity only updates at discrete intervals (once per year, once per generation), while continuous growth assumes it's updating constantly at every instant. For the same nominal rate, continuous growth always produces a slightly higher final value than compound growth, since e^r is always slightly greater than (1+r) for positive r.

It gives an intuitive, memorable sense of how fast something is really growing — "doubles every 14 years" is often easier to reason about at a glance than a raw percentage growth rate, and it's the basis of the well-known "rule of 70" approximation (doubling time ≈ 70/growth rate percentage) used for quick mental estimates.

No — real populations, cultures, and outbreaks eventually hit resource limits, saturation, or other constraints that slow growth down, which is why more realistic long-term models (like logistic growth) eventually replace pure exponential growth once those limits start to matter.

The same formulas still apply, but they describe exponential decay instead of growth — the quantity shrinks by a fixed percentage each period rather than growing, and the "doubling time" calculation would instead need to be reinterpreted as a halving time.

e is defined as the limit of (1 + r/n)^(n) as n (the number of compounding periods per unit time) grows toward infinity — it's precisely the mathematical constant that emerges when you compound growth infinitely often, which is exactly what "continuous" growth means.