When is exponential growth appropriate?

Exponential growth approximates real populations only when resources are unlimited and per-capita birth and death rates are constant, for example bacteria in fresh nutrient broth or invasive species in early colonisation.

What's the difference between r and λ?

r is the continuous intrinsic growth rate used in N = N₀·e^(rt). λ is the discrete finite rate of increase used in N_(t+1) = λ·N_t. They're related by λ = e^r.

What happens when r < 0?

The equation describes exponential decay. Useful for modelling population decline or radioactive-style first-order decay.

Why log natural and not log base 10?

The model is built on the continuous solution to dN/dt = rN, whose solution uses the natural log. To rearrange for t or r you take ln, not log₁₀.

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Exponential population growth

Q: What happens when r < 0?

The equation describes exponential decay. Useful for modelling population decline or radioactive-style first-order decay.

Q: Why log natural and not log base 10?

The model is built on the continuous solution to dN/dt = rN, whose solution uses the natural log. To rearrange for t or r you take ln, not log₁₀.

Model continuous exponential growth with N(t) = N₀ · e^(rt). Solve for N(t), t, r, or N₀ by picking the unknown.

Formula

N(t) = N₀ · e^(rt)

Rearranged:

t = ln(N / N₀) / r
r = ln(N / N₀) / t
N₀ = N / e^(rt)

Inputs

Solve forN₀ (initial population)r (intrinsic growth rate, per unit time)Use the same time unit as t.t (elapsed time)

Continuous exponential growth: N(t) = N₀ · e^(rt). For r < 0 the model describes exponential decay.

Result

Final population N(t)

271.8282

N = 100 · e^(0.05 × 20) = 271.83

Worked example

A bacterial culture starts at N₀ = 100 cells with intrinsic growth rate r = 0.05 per minute. What is the population after t = 20 minutes?

N(20) = 100 · e^(0.05 × 20) = 100 · e¹ = 100 · 2.71828 ≈ 272 cells.

Going the other way: how long until the population reaches 1000? t = ln(1000/100) / 0.05 = ln(10) / 0.05 ≈ 46 minutes.

How this calculator works

The calculator picks one of four rearrangements of N = N₀·e^(rt) based on which variable you choose as the unknown, then evaluates it numerically using natural log and exp. For r < 0 the equation models exponential decay.

Common questions

When is exponential growth appropriate?: Exponential growth approximates real populations only when resources are unlimited and per-capita birth and death rates are constant, for example bacteria in fresh nutrient broth or invasive species in early colonisation.
What's the difference between r and λ?: r is the continuous intrinsic growth rate used in N = N₀·e^(rt). λ is the discrete finite rate of increase used in N_(t+1) = λ·N_t. They're related by λ = e^r.
What happens when r < 0?: The equation describes exponential decay. Useful for modelling population decline or radioactive-style first-order decay.
Why log natural and not log base 10?: The model is built on the continuous solution to dN/dt = rN, whose solution uses the natural log. To rearrange for t or r you take ln, not log₁₀.