What does r represent in the exponential growth formula?

r is the intrinsic (per-capita) growth rate, expressed per unit of time. For bacteria dividing every 20 minutes, r = ln(2) / (20/60 hours) ≈ 2.08 per hour. A negative r models exponential decay.

What units should I use?

Time and growth rate must use the same unit. If r is per hour, t must be in hours. The result N(t) will be in the same unit as N0 (individuals, cells, CFU, etc.).

How is doubling time calculated?

Doubling time Td = ln(2) / r ≈ 0.693 / r. This is the time required for the population to double. It only applies when r > 0.

What is the difference between exponential and logistic growth?

Exponential growth assumes unlimited resources — the population grows without bound. Logistic growth adds a carrying capacity K that slows growth as the population approaches K. This calculator models the exponential phase only.

Can I use this for radioactive decay?

Yes. Enter a negative r equal to the decay constant (−λ). The formula N(t) = N0 × e^(−λt) is mathematically identical to exponential decay.

What if my growth rate is a percentage?

Convert the percentage to a decimal first (e.g., 5% per year → r = 0.05). Note that a continuous growth rate r is slightly different from a discrete percentage growth — r = ln(1 + percentage_rate) for exact conversion.

Exponential Growth Biology Calculator

Name: Exponential Growth Biology Calculator
Availability: InStock
Author: Nham Vu

Enter initial population, intrinsic growth rate, and time to compute final population size using the N(t) = N0 * e^(r*t) formula.

N(t) = N₀ × e^r t

Fill in three fields — the calculator solves for the fourth.

Initial Population N₀

Growth Rate r (per unit time; negative = decay)

Time t

Final Population N(t) (leave blank to solve)

Enter at least three values and click Calculate

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Summary

Enter initial population, intrinsic growth rate, and time to compute final population size using the N(t) = N0 * e^(r*t) formula.

How it works

Enter the initial population size (N0) — the starting count of organisms or cells.
Enter the intrinsic growth rate (r) — a positive value means growth, negative means decay.
Enter the time elapsed (t) — use consistent units (hours, days, years) with your r value.
The calculator applies N(t) = N0 × e^(r×t) and returns the final population.
The doubling time (ln(2)/r) is shown automatically when r > 0.
A step-by-step breakdown shows each part of the calculation.

Use cases

Predict bacterial colony size after a given incubation period.
Model cell culture growth in a bioreactor experiment.
Estimate wildlife population changes over time for conservation planning.
Calculate radioactive decay using a negative growth rate.
Verify homework answers for biology or ecology courses.
Simulate unconstrained population growth before applying a logistic model.
Estimate COVID-19 or other pathogen spread during early exponential phase.
Compute doubling time for any exponentially growing system.

Frequently Asked Questions

Last updated: 2026-07-01 · Reviewed by Nham Vu