Continuous Growth Rate Formula: Derivation and Use

The formula at a glance

For a quantity V₀ growing continuously at rate r, the value at time t is V(t) = V₀ × e^(rt). Take the natural log of both sides and you get ln(V/V₀) = rt, which makes the rate r the slope of value on a log scale. That is the entire model.

Why the constant e shows up

The number e is the limit of (1 + 1/n)^n as n approaches infinity. It is what compound interest looks like when the compounding interval shrinks to zero. Every continuous exponential model has e as its base because e is the natural choice when growth is proportional to the current size and the rate applies at every instant rather than at discrete ticks.

From continuous growth to doubling time

Set V(t) equal to 2V₀ and solve for t. The starting amount cancels, leaving e^(rt) = 2. Taking the natural log of both sides gives rt = ln(2), so t = ln(2) / r. That is the continuous doubling time formula in its simplest form. For the discrete version, see The Doubling Time Formula.

When to choose continuous

Use continuous when the underlying process is genuinely smooth in time. Examples: bacterial culture growth, radioactive decay, theoretical instantaneous compounding in finance, or any biological process running on a continuous schedule. Use discrete when growth posts in fixed steps: annual interest, monthly user growth, quarterly dividends. At low rates the two formulas agree to within a fraction of a percent. The gap grows as rates climb.

Worked example

A bacterial culture grows at a continuous rate of 2.08 per hour. The doubling time is t = ln(2) / 2.08 = 0.693 / 2.08 = 0.333 hours, or about 20 minutes. That matches the textbook generation time for E. coli in ideal conditions. Drop the rate into the calculator's continuous mode and the graph shows the curve doubling once between 0 and roughly 0.33 hours.

For more biology-flavored scenarios, see Population Doubling Time. For finance-flavored ones, see Doubling Time in Finance.