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Exponential Growth Calculator

This exponential growth calculator solves x(t) = x₀·(1 + r)^t for the final value, the rate or the time. Example: 1,000 growing 5 % per period reaches 1,628.89 after 10 periods; the doubling time is about 14.2 periods.

Example: What do you want to find: Final value x(t) · Starting value x₀ 1,000 · Rate per period 5 % · Number of periods t 10 → Result: Final value x(t) = 1,628.89 (start 1,000, after 10 periods). Source: Wikipedia – Exponentielles Wachstum. Updated: .

%
Negative for decay, above −100.
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Result

Result
Final value x(t) = 1,628.89 (start 1,000, after 10 periods)
All values
QuantityValue
Starting value x₀1,000
Rate r per period5 %
Periods t10
Final value x(t)1,628.89
Total growth factor1.62889
Total change62.89 %
Continuous rate k = ln(1 + r)0.0487902
Doubling time14.2067
Calculation steps
  • x(t) = x₀ · (1 + r)^t = 1,000 · 1.05^10 = 1,628.89.
  • Continuous rate k = ln(1 + r) = ln(1.05) = 0.0487902; equivalently x(t) = x₀ · e^(k·t).
  • Doubling time = ln 2 / ln(1.05) = 14.2067 periods.
The rate applies per period (year, day, hour …) and is assumed constant. Real processes are often exponential only for a while.

How it is calculated

Exponential growth means a quantity increases by the same percentage in every period. It therefore speeds up, because the increase itself grows – as with compound interest, bacteria cultures or virus counts at the start of a wave.

Formula and rearrangements

The model is x(t) = x₀ · (1 + r)^t with starting value x₀, rate r (as a decimal) and time t. The final value comes straight from it. The rate follows from r = (x(t) / x₀)^(1/t) − 1 and the time from t = ln(x(t) / x₀) / ln(1 + r). The doubling time is ln 2 / ln(1 + r) and does not depend on the starting value. With a negative rate the same formula describes exponential decay with a half-life.

Three worked cases

Final value: 1,000 at 5 % over 10 periods gives 1,000 · 1.05¹⁰ ≈ 1,628.89. Rate: to double from 100 to 200 in 5 periods you need 2^(1/5) − 1 ≈ 14.87 % per period. Time: from 100 to 200 at 7 % takes ln 2 / ln 1.07 ≈ 10.24 periods.

Frequently asked questions

How do I calculate exponential growth?

Use x(t) = x₀ · (1 + r)^t. Insert the starting value, the rate per period as a decimal and the number of periods. With 1,000, 5 % and 10 periods the result is about 1,628.89.

How do I find the growth rate from two values?

Divide the final value by the starting value, take the t-th root and subtract 1. From 100 to 200 in 5 periods that is about 14.87 % per period.

How does the continuous rate differ?

The continuous rate k = ln(1 + r) belongs to the form x₀ · e^(k·t). Both forms give the same values; k is smaller than r, about 0.0488 instead of 5 %.

Sources and legal basis

As of: · Editorial policy

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