Exponential Growth Calculator

The Exponential Growth Calculator solves the continuous exponential growth equation P(t) = P₀ · e^(rt) for whichever variable you don't already know: the final amount, the initial amount, the growth rate, or the time elapsed. Because the equation is symmetric, the same tool also handles continuous decay — just enter a negative growth rate.

It's built for students working through exponential-growth problems, anyone modeling population growth, investment growth, radioactive decay, or bacterial cultures, and for checking homework or spreadsheet formulas against a step-by-step derivation.

How to use it

1. Pick what you want to Solve For — Final Amount, Initial Amount, Growth Rate, or Time Elapsed.
2. Fill in the other three fields. The field matching your solve-for choice disappears since that's the value being computed.
3. When you're not solving for the rate itself, switch Enter Rate As to Half-Life to specify a decay problem the way it's usually given (e.g. "half-life of 5,730 years") instead of converting it to a percentage rate by hand.
4. Read the Result table for the solved value plus the full set of variables, a growth/decay classification, and a doubling-time (or half-life) figure.
5. Check Step-by-Step Solution for the formula substitution that produced the answer.

Results update automatically as you type. For example, solving for the final amount with P₀ = 1000, r = 5%, t = 10 returns P(t) = 1648.72.

Working directly from a half-life

Half-life problems (radioactive decay, drug elimination, carbon dating) are almost always stated as a half-life, not a percentage rate. Set Enter Rate As to Half-Life and type the half-life directly — the tool converts it internally via r = -ln(2) / half-life before solving, so you never have to do that conversion yourself. This only applies when you're not solving for the rate: if you want to find a half-life from known amounts and time, solve for Growth Rate instead and read it off the Half-Life row in the results.

What is continuous exponential growth?

Continuous exponential growth describes a quantity that grows (or shrinks) at a rate proportional to its current size at every instant, not just once per discrete period. That's what the e^(rt) term captures — e is Euler's number (≈ 2.71828), r is the continuous growth rate, and t is the elapsed time. It's the standard model for population growth, continuously compounded interest, and radioactive decay.

Why does a negative rate work for decay?

The formula P(t) = P₀ · e^(rt) doesn't change shape when r is negative — it just means the exponent rt is negative, so e^(rt) is between 0 and 1 and the amount shrinks over time instead of growing. The calculator reports this case as Decay in the Growth Type row and shows a Half-Life instead of a Doubling Time.

What is doubling time / half-life?

• Doubling Time (shown when the rate is positive) is how long it takes the amount to double: ln(2) / r.
• Half-Life (shown when the rate is negative) is how long it takes the amount to halve: ln(2) / |r|.

Both come from the same identity — solving P(t) = 2·P₀ (or P₀/2) for t in the growth equation.

Why must the amounts be positive?

The Initial Amount and Final Amount represent physical quantities — population, balance, mass — that the model treats as strictly positive, and solving for the rate or time takes ln(P(t) / P₀), which is only defined for a positive ratio. Enter magnitudes only; use the sign of the growth rate to express growth vs. decay.

Privacy

This calculator runs entirely in your browser. Your numbers are never uploaded, logged, or stored — the computation happens locally on your device.