Exponential Growth Rate Calculator
Exponential growth describes a quantity that increases in proportion to its current size. The continuous growth rate constant captures how fast this happens per unit time. This calculator finds that constant from an initial value, a final value and the elapsed time, then reports the equivalent continuous percentage rate and the doubling time. Enter the values to characterise the growth.
Exponential growth rate formula
model: N(t) = N0 * e^(r * t)
growth rate r = ln(final / initial) / t
doubling time = ln(2) / r
The natural logarithm of the fold change, divided by the elapsed time, gives the continuous growth rate. Doubling time follows from the rate using the natural log of 2.
Worked example
A value grows from 100 to 200 over 10 units of time. The fold change is 200 / 100 = 2, and ln(2) = 0.693147. So r = 0.693147 / 10 = 0.069315 per unit time, or 6.93 percent continuous. The doubling time is ln(2) / r = 0.693147 / 0.069315 = 10.00, which matches the period over which it doubled.
Exponential growth rate: frequently asked questions
What is the exponential growth rate constant?
The exponential growth rate constant, written r or k, is the continuous per-unit-time rate at which a quantity grows under exponential growth. In the model N(t) = N0 times e to the power of (r t), r is the instantaneous fractional growth per unit time. A positive r indicates growth, a negative r indicates decay. It is the natural-logarithm-based counterpart of a periodic percentage growth rate.
How do you calculate the growth rate from two values?
Rearranging the exponential model gives r = ln(final / initial) divided by the elapsed time, where ln is the natural logarithm. This finds the continuous rate that takes the initial value to the final value over the given period. For example, doubling over 10 units of time gives r = ln(2) / 10 = 0.0693 per unit time.
How is doubling time related to the growth rate?
For continuous exponential growth, the doubling time is ln(2) divided by r, approximately 0.693 / r. A larger growth rate means a shorter doubling time. This is the exact basis of the well known 'rule of 70', which estimates doubling time as 70 divided by the percentage growth rate, since ln(2) is about 0.70.
When is the exponential model appropriate?
Exponential growth applies when a quantity grows in proportion to its current size and resources are effectively unlimited, such as early bacterial growth, compound interest, or an unconstrained population. When resources become limiting, growth slows and a logistic model fits better. Use the exponential growth rate for the unconstrained phase or for short periods.
Official sources
- National Institute of Standards and Technology: NIST Digital Library of Mathematical Functions (exponential function).
- U.S. Census Bureau: Census growth-rate methodology.
Reviewed by the CalculatorHub team, edited by James Graham, 19 June 2026. See our methodology.