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- Works out exponential growth or decay.
- Finds the final amount, the rate, the time needed or the starting amount.
- Uses the discrete model x₀(1 ± r)^t or the continuous model x₀e^(±rt).
- Shows the doubling time or half-life and a table of values.
How to use the exponential growth calculator
Choose what to find
Final amount, rate, time or starting amount.
Choose growth or decay
Growth increases each period; decay decreases.
Enter the known values
Starting amount, rate per period, number of periods or final amount.
Pick the model
Discrete (per period) or continuous.
Exponential growth and decay formulas
Exponential change happens when a quantity grows or shrinks by the same percentage each period, such as populations, investments, bacteria or radioactive material. The bigger the amount, the bigger the change.
Formulas
Growth: x(t) = x₀ × (1 + r)^t
Decay: x(t) = x₀ × (1 − r)^t
Continuous: x(t) = x₀ × e^(±rt)
Doubling time = ln 2 ÷ ln(1 + r)
Example 1: 1,000 growing 5% for 10 periods
Discrete growth.
- Final amount 1,628.89
- Doubling time 14.21 periods
- Continuous model instead 1,648.72
Example 2: 1,000 decaying 5% for 10 periods
Discrete decay.
- Final amount 598.74
- Half-life 13.51 periods
Example 3: doubling in 10 periods
From 1,000 to 2,000.
- Rate needed 7.18% per period
Where exponential models are used
Population growth
Model cities, bacteria or animal populations.
Finance
Compound growth of money and inflation.
Radioactive decay
Half-lives in physics and chemistry.
Depreciation
Value lost each year by cars or equipment.
For money with deposits, use the compound interest calculator. Find a yearly growth rate with the CAGR calculator, or estimate doubling time with the rule of 72 calculator.
Exponential growth calculator FAQ
What is the exponential growth formula?
x(t) = x₀ × (1 + r)^t, where x₀ is the start, r the rate per period and t the number of periods.
What is the difference between growth and decay?
Growth multiplies by (1 + r) each period; decay multiplies by (1 − r).
What is continuous growth?
Growth that compounds constantly, modelled with e: x(t) = x₀e^(rt).
How do you find the doubling time?
Divide ln 2 by ln(1 + r). At 5% per period it is about 14.21 periods.
How do you find the half-life?
For decay, divide ln 2 by −ln(1 − r). At 5% per period it is about 13.51 periods.
Can the rate be per year, month or day?
Yes. Use any period, as long as the rate and the number of periods use the same one.