# Exponential Population Growth (N_t = N₀·e^(rt)) Compute N_t = N₀·e^(rt) with fold change, doublings, doubling time, or half-life. > Canonical page: https://elysiatools.com/en/tools/exponential-growth-population - **Category:** Health - **Keywords:** exponential growth, population growth, n0e rt, doubling time, half life, growth rate, bacterial growth, compound growth, ecology calculator, microbiology, lab calculator, math biology ## Overview Compute future population size with the continuous exponential model N_t = N₀ · e^(rt). Enter starting size, constant per-capita rate r and elapsed time t (same units) to obtain N_t, fold change, number of doublings, and either doubling time (r > 0) or half-life (r < 0). The calculator is intended for unlimited-resource phases such as bacterial log growth. ## Inputs - **Initial population (N₀)** (number): Population size at t = 0 (cells, individuals, biomass — any unit). - **Growth rate r (per time unit)** (number): Constant per-capita rate: r > 0 growth, r < 0 decline, r = 0 stationary. - **Elapsed time (t)** (number): Time elapsed in the same unit as r. - **Decimal places** (number) ## When to use - Projecting cell or bacterial counts during unrestricted log-phase growth in the lab. - Estimating how a population multiplies or declines at a constant per-capita rate over a known interval. - Deriving doubling time or half-life directly from a measured growth or decline rate r. ## How it works - Enter the initial population N₀ (any consistent unit: cells, individuals or biomass). - Supply the constant rate r using the same time unit as t; positive r produces growth, negative r produces decline. - Specify the elapsed time t. - The tool returns N_t = N₀ · e^(rt) together with fold change, doublings and the appropriate characteristic time (doubling time or half-life). ## Use cases - Forecasting colony-forming units after a defined incubation period in a microbiology laboratory. - Simple unbounded population projections in early-stage ecological or teaching models. - Calculating half-life of a declining population or decaying biomass. ## Frequently asked questions ### What formula is used? N_t = N₀ · e^(rt), the standard continuous exponential-growth equation. ### Must r and t use the same time unit? Yes. Both must be expressed in identical units (hours, days, etc.). ### What if the growth rate r is negative? The population declines exponentially; the calculator reports half-life instead of doubling time. ### How are fold change and doublings obtained? Fold change equals e^(rt); the number of doublings is rt / ln(2). ### When should I not use this model? When resources become limiting or the population approaches carrying capacity; a logistic model is then more appropriate. ## Related tools - [A-a Gradient (Alveolar-Arterial O₂)](https://elysiatools.com/en/tools/a-a-gradient-calculator): Calculate the alveolar-arterial O₂ gradient to distinguish hypoxemia causes. PAO₂ = FiO₂×(P_atm−47) − PaCO₂/0.8; A-a = PAO₂ − PaO₂. Normal ≈ (Age+10)/4. Elevated → V/Q mismatch, shunt, diffusion impairment. Derived from StatPearls, LITFL, Cornell PICU, and Martin 1999. 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