Exponential Growth Calculator (growth & decay, doubling time, half-life)

Understanding the exponential growth calculator — concept and principle

This calculator projects a quantity that grows or shrinks by a constant percentage per period. It gives the value after t periods, the change, the total growth factor, the doubling time or half-life and the equivalent continuous rate. Use it for savings, populations, inflation, bacteria and radioactive decay.

Each period adds 5 % of the current amount, so the bars grow faster over time.

How it is calculated

Once per period: x(t) = x₀(1 + r)ᵗ

Continuous: x(t) = x₀·e^(kt), with k = ln(1 + r) for the same curve

Doubling time = ln 2 ÷ k; half-life (negative rate) = −ln 2 ÷ k

Total growth factor = x(t) ÷ x₀

Enter a negative rate for decay; t does not have to be a whole number.

The principle

Exponential growth adds a fixed percentage of the current amount each period, so the additions themselves grow: the bigger it gets, the faster it grows in absolute terms. Multiplying by (1 + r) t times gives the formula. Continuous growth is the limit of compounding more and more often, which leads to e. Because the factor per period is constant, the time to double is the same at any size.

Worked example

Default inputs: start 1,000, 5 % per period, 10 periods, applied once per period.

Growth factor = 1.05¹⁰ = 1.628894627.
Value after 10 periods = 1,000 × 1.628894627 = 1,628.894627.
Change = 1,628.89 − 1,000 = 628.8946268.
Continuous rate k = ln 1.05 = 4.879016417 %; equivalent rate per period 5 %.
Doubling time = ln 2 ÷ ln 1.05 = 14.20669908 periods; half-life — (no decay).

Each point is the calculator re-run with only “Starting value” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Starting value	Value after t periods
250	407.2236567
500	814.4473134
750	1,221.67097
1,000	1,628.894627
1,250	2,036.118283
1,500	2,443.34194
2,000	3,257.789254

How to read the result

Over 10 periods the total gain, about 62.9 %, is well above 10 × 5 % = 50 %, because each period's growth builds on the last. The gap widens quickly with time: at the same rate the value doubles in about 14.2 periods and quadruples in about 28.4.

Common mistakes

Adding 5 % of the starting value each period (that is linear growth).
Mixing yearly rates with monthly periods.
Entering 5 % as 0.05 when the field expects percent, or the reverse.
Expecting real growth to stay exponential indefinitely.

Keep in mind: This is a model with a constant rate. Real returns, populations and prices change over time; the result is not a forecast or financial advice.

Frequently asked questions

What is the formula for exponential growth?

x(t) = x₀(1 + r)ᵗ, or x₀·e^(kt) for continuous growth.

How long does it take to double at 5 %?

About 14.2 periods.

How do I model decay?

Enter a negative rate; the page then gives the half-life.

Sources: OpenStax, College Algebra 2e (2021), §6.7 Exponential and Logarithmic Models

Checked 9 Nov 2026. OpenStax, Algebra and Trigonometry 2e, Chapter 6 Key Equations (exponential growth f(x) = a·bˣ; continuous growth A(t) = a·eʳᵗ; doubling time t = ln 2 / k; half-life t = −ln 2 / k), §6.1 and §6.7.

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