Exponential Growth & Decay Calculator (Free)

An exponential growth calculator uses x(t) = x₀(1 + r)^t, where x₀ is the starting amount, r the rate per time step and t the number of steps. For example, 1,000 growing 5% a year for 10 years becomes 1,000 × 1.05¹⁰ ≈ 1,628.89. A negative rate gives exponential decay.

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How to use the exponential growth calculator

1. Pick what you want to find: the final value x(t), the rate, the time, the initial value x₀, or the result of a doubling time or half-life.
2. Choose the model: periodic, x₀(1 + r)^t, or continuous, x₀e^kt.
3. Enter the other values. Use a negative rate, such as −10, for decay, and pick the time unit your rate uses.
4. Read the answer, then open Show steps, the graph and the table of values. The example buttons load a population, bacteria, carbon-14 and car-value problem.

Exponential growth and decay formulas

Periodic model: x(t) = x₀(1 + r)^t

• x₀ is the initial value, at t = 0.
• r is the rate per time step as a decimal: 5% growth is 0.05, 10% decay is −0.10.
• t is the time, in the same unit as the rate.
• 1 + r is the growth factor: above 1 means growth, between 0 and 1 means decay.

Continuous model: x(t) = x₀e^kt, where k is the continuous rate and e ≈ 2.71828. The two models describe the same curve when k = ln(1 + r). A 5% periodic rate equals a continuous rate of about 4.879%, and a continuous rate of 5% equals a periodic rate of about 5.1271%.

Worked example: periodic vs continuous growth

Start with 1,000 and a rate of 5% for 10 time units. Periodic: 1,000 × 1.05¹⁰ ≈ 1,628.89. Continuous: 1,000 × e^{0.05 × 10} = 1,000 × e^0.5 ≈ 1,648.72. The same percentage gives a larger result in the continuous model, because the growth is added at every instant.

Worked example: exponential decay

1,000 decreasing 10% per period for 3 periods: 1,000 × 0.9³ = 729. The total drop is 27.1%, not 30%, because each period takes 10% of what is left, not of the original amount.

How to solve for the rate, time or initial value

• Rate: r = (x(t) ÷ x₀)^1/t − 1, or k = ln(x(t) ÷ x₀) ÷ t. Growing from 1,000 to 2,000 in 10 years takes r ≈ 7.1773% a year.
• Time: t = ln(x(t) ÷ x₀) ÷ ln(1 + r), or ln(x(t) ÷ x₀) ÷ k. At 5% a year, 1,000 reaches 2,000 after about 14.2067 years.
• Initial value: x₀ = x(t) ÷ (1 + r)^t. Divide out the growth to go back in time.

Doubling time and half-life

The doubling time is how long a growing amount takes to double: t₂ = ln 2 ÷ ln(1 + r), or ln 2 ÷ k. The half-life is how long a decaying amount takes to halve: t_½ = ln 2 ÷ |ln(1 + r)|. For a quick estimate, divide 70 (or 72) by the percentage rate.

When you know the doubling time or half-life instead of the rate, use x(t) = x₀ × 2^t/T or x(t) = x₀ × (1/2)^t/T, where T is the doubling time or half-life.

Real-world examples

Population growth

A town of 50,000 grows 2% a year. After 10 years: 50,000 × 1.02¹⁰ ≈ 60,949.72, so about 60,950 people. At that rate the population doubles every 35.00 years.

Bacteria doubling

A culture starts with 100 bacteria and doubles every 20 minutes. Three hours is 180 ÷ 20 = 9 doublings, so the count is 100 × 2⁹ = 51,200.

Radioactive decay: carbon-14

Carbon-14 has a half-life of 5,730 years (NOAA). After 10,000 years, a 100-gram sample has passed 10,000 ÷ 5,730 ≈ 1.745 half-lives, leaving 100 × (1/2)^1.745 ≈ 29.83 grams. As a continuous rate, k = −ln 2 ÷ 5,730 ≈ −0.000121 per year.

Depreciation

A car bought for $30,000 that loses 15% of its value each year is worth 30,000 × 0.85⁵ ≈ $13,311.16 after 5 years. Money growing at a fixed interest rate follows the same model; see the compound interest calculator.

Exponential vs linear growth

Linear growth adds the same amount each step; exponential growth multiplies by the same factor. Starting at 1,000, adding 50 a year gives 1,500 after 10 years and 2,500 after 30. Growing 5% a year gives 1,628.89 after 10 years and about 4,321.94 after 30. The gap keeps widening because each step’s growth is based on a bigger amount.

Common mistakes

• Entering 5% as 5 instead of 0.05 in the formula (the calculator takes 5 and converts it for you).
• Using a yearly rate with time in months. Match the units, or convert the rate first.
• Treating decay as subtraction: 10% decay for 3 periods leaves 72.9%, not 70%.
• Mixing up r and k: x₀(1 + r)^t and x₀e^kt agree only when k = ln(1 + r).
• Assuming the rate stays constant forever. Real populations level off and prices change; long projections are what the model says, not a forecast.

Before you calculate, predict the direction: a positive rate should give a bigger answer, a negative rate a smaller one. Then state the model, substitute, and write the units with your answer.

Sources

Checked November 2026.

• NOAA Global Monitoring Laboratory, Radioactive decay (carbon-14 half-life of 5,730 years)
• National Institute of Justice, What is carbon dating?

Exponential growth and decay practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

How do you calculate exponential growth?

Multiply the starting amount by the growth factor raised to the time: x(t) = x₀(1 + r)^t. For 200 growing 3% a year for 5 years, 200 × 1.03⁵ ≈ 231.85.

What is the difference between continuous and periodic growth?

Periodic growth adds the percentage once per step; continuous growth adds it at every instant, using e^kt. For the same percentage, continuous growth ends slightly higher: 1,648.72 against 1,628.89 for 1,000 at 5% for 10 steps.

How do I model exponential decay?

Use a negative rate. A 10% decrease per period means r = −0.10, so the factor is 0.9 and x(t) = x₀ × 0.9^t. With a half-life T, use x(t) = x₀ × (1/2)^t/T.

What is doubling time?

The time an amount takes to double at a constant growth rate: ln 2 ÷ ln(1 + r). At 5% per year it is about 14.21 years. The half-life is the same idea for decay.

How do you find the growth rate from two values?

Divide the later value by the earlier one, take the t-th root and subtract 1: r = (x(t) ÷ x₀)^1/t − 1. Use the Rate tab to see each step.

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