About the Exponential Growth and Decay Calculator
This exponential growth and decay calculator projects how a quantity changes when it grows or shrinks by a constant percentage each period. Enter a starting value, a rate per period (use a negative rate for decay) and the number of periods to get the final value, the total change, the growth factor, and the doubling time or half-life.
It fits population growth, bacteria cultures, viral spread, compound interest, inflation, depreciation, radioactive decay and drug elimination. Choose discrete growth, x(t) = x₀(1 + r)ᵗ, when the change is applied once per period, or continuous growth, x(t) = x₀eʳᵗ, when it happens smoothly all the time, as in many natural processes.
Time can be in any unit — years, hours, generations — as long as the rate is per the same unit. The chart shows the curve over time so you can see how growth accelerates or decay levels off.
With the default inputs, the final value x(t) is 1,628.8946. Change any value above to recalculate instantly.
How to use the exponential growth and decay calculator
- 1Enter the starting value x₀.
- 2Enter the rate per period — positive for growth, negative for decay.
- 3Enter how many periods to project.
- 4Choose discrete or continuous growth.
- 5Read the final value, doubling time or half-life, and the curve on the chart.
Formula and method
In discrete growth the rate r is applied once per period, so after t periods the starting value is multiplied by (1 + r)ᵗ. In continuous growth the change compounds instantly, giving x₀eʳᵗ, which is always slightly larger for positive r. A negative r gives decay in both models.
Doubling time is how long it takes to multiply by 2: ln 2 ÷ ln(1 + r) for discrete growth and ln 2 ÷ r for continuous growth. For decay the same formulas with |r| give the half-life. The Rule of 72 (72 ÷ rate %) is a quick mental approximation of the discrete doubling time.
- x₀
- Initial value at t = 0
- r
- Growth rate per period as a decimal (negative for decay)
- t
- Number of periods elapsed
- e
- Euler’s number ≈ 2.71828
Worked examples
$1,000 growing 5% a year for 10 years
1,000 × 1.05¹⁰ = 1,628.89, an increase of about 62.9%. At 5% per period the value doubles every ln 2 ÷ ln 1.05 ≈ 14.2 periods.
Bacteria growing continuously at 20% per hour for 12 hours
500 × e^(0.2 × 12) = 500 × e^2.4 ≈ 5,511.6 bacteria — about 11 times the starting count. The doubling time is ln 2 ÷ 0.2 ≈ 3.47 hours.
Car value falling 15% a year for 5 years
30,000 × 0.85⁵ ≈ 13,311, so the car loses about 55.6% of its value. At 15% per year the value halves roughly every 4.27 years.
Frequently asked questions
What is the exponential growth formula?+
For growth applied once per period, x(t) = x₀(1 + r)ᵗ, where x₀ is the start value, r the rate as a decimal and t the number of periods. For continuous growth use x(t) = x₀eʳᵗ.
What is the difference between exponential growth and decay?+
Both change by a constant percentage each period. Growth has a positive rate and the quantity accelerates upward; decay has a negative rate and the quantity shrinks toward zero but never quite reaches it.
How do I calculate doubling time?+
Doubling time = ln 2 ÷ ln(1 + r) for discrete growth, or ln 2 ÷ r for continuous growth. A quick estimate is the Rule of 72: divide 72 by the percentage rate, so 6% growth doubles in about 12 periods.
When should I use continuous instead of discrete growth?+
Use continuous growth when change happens smoothly all the time, such as population biology, radioactive decay or continuously compounded interest. Use discrete growth when the change is applied at set intervals, like annual interest or yearly depreciation.
How do I find the growth rate from two values?+
Rearrange the formula: r = (x(t) ÷ x₀)^(1/t) − 1 for discrete growth, or r = ln(x(t) ÷ x₀) ÷ t for continuous growth. The CAGR calculator does this for investment returns.