This exponential growth calculator projects how a population, culture or colony grows when each member reproduces at a constant rate: starting amount, growth rate and time in — final amount, total growth and doubling time out.
Model: N(t) = N₀ · e^(r·t), the continuous exponential. A negative rate computes exponential decay instead — the halving time then replaces doubling time.
The Math of Things That Multiply
Exponential growth happens when the growth itself grows — bacteria dividing, rabbits breeding, infections spreading, money compounding (the same equation powers our Compound Interest Calculator, where the “population” is your balance). The continuous model: N(t) = N₀ · e^(rt). Its signature constant is the doubling time, ln(2) ÷ r — fixed no matter how big the population gets, which is exactly what makes exponentials deceptive: they look slow, slow, slow… then sudden. This calculator is part of our Biology Calculators collection.
Worked Example (Verified)
A culture of 100 bacteria growing at 30% per hour:
- After 5 hours: N = 100 × e^(0.3×5) = 100 × e^1.5 = ≈ 448 bacteria
- Doubling time: ln(2) ÷ 0.3 = 2.31 hours — check: 100 → 200 → 400 in 4.62 hours, and the model says 448 at 5 hours ✔
- After 24 hours the same math gives ≈ 134,000 — from a hundred cells in a day. This is why food safety rules treat the “danger zone” in hours, not days.
When Exponentials Stop (They Always Do)
- Real growth hits limits — food, space, waste buildup — and bends into an S-shaped logistic curve approaching a carrying capacity. Use exponential projections for the early phase only.
- Percentage points vs. rates: a rate quoted “per hour” must be paired with hours; convert rates before mixing time units — the most common student error this calculator cannot catch for you.
- Decay is the mirror image: negative r gives half-lives — radioactive decay and drug clearance follow the halving schedule this tool reports when r < 0.
Related Tools
- Compound Interest Calculator — exponential money
- Half-Life Calculator — exponential decay, dedicated
- Biology Calculators — the full collection
Frequently Asked Questions
What is the exponential growth formula?
N(t) = N₀ · e^(rt): starting amount N₀, continuous growth rate r (as a decimal), time t. After one doubling time (ln 2 ÷ r), the amount exactly doubles.
How do I find the doubling time?
Doubling time = ln(2) ÷ r ≈ 0.693 ÷ r. At 30% growth per hour: 2.31 hours. For discrete annual growth, the Rule of 70 gives the handy approximation 70 ÷ (percent rate).
What is the difference between exponential and linear growth?
Linear adds a constant amount per period; exponential adds a constant percentage — so its additions grow. 100 growing linearly at +30/period reaches 250 in 5 periods; at 30%/period it reaches about 448.
Can this calculator do exponential decay?
Yes — enter a negative rate (e.g. −10). The “doubling time” then reads as the halving time.
Why does my textbook use N = N₀(1 + r)^t instead?
That is the discrete (per-period compounding) version; e^(rt) is the continuous limit. For small rates they nearly agree; for large rates or long times the continuous version grows slightly faster. Microbiology and pharmacology mostly use the continuous form shown here.
Is this exponential growth calculator free?
Yes — free, instant, no sign-up.