Exponential Growth Calculator
Solve continuous exponential growth (or decay) equations — P(t) = P₀ · e^(rt) — for the final amount, initial amount, growth rate, or time elapsed, with a doubling-time/half-life figure and a full step-by-step derivation.
Input
Negative values model continuous decay instead of growth.
Output
| Metric | Value |
|---|---|
| No data yet | |
Guides
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
- Pick what you want to Solve For — Final Amount, Initial Amount, Growth Rate, or Time Elapsed.
- Fill in the other three fields. The field matching your solve-for choice disappears since that's the value being computed.
- 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.
- 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.
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.
Use it from code
From 3 credits per callREST API
curl -X POST https://api.iotools.cloud/v1/tool/exponential-growth-calculator \
-H "Authorization: Bearer YOUR_API_KEY" \
-H "Content-Type: application/json" \
-d '{
"solveFor": "final",
"initialAmount": "1000",
"growthRate": "5",
"timeValue": "10"
}'Swap in your own key from your account. The tool's fields are the body — no wrapper.
Ask an AI agent
Use the IOTools `exponential-growth-calculator` tool (Exponential Growth Calculator) on this input:
YOUR_INPUT_HEREPaste this at any agent connected to the IOTools MCP server, then add your input.