Exponential Growth and Doubling Time: Bacteria, Populations, and Compounding
October 29, 2026
Exponential growth occurs when a quantity increases by a constant percentage in each period, so the amount added grows as the total grows. The formula is N = N₀ × (1 + r)^t for a rate r per period, or N = N₀ × 2^(t/T) when the doubling time T is known.
Bacteria that double every 20 minutes start with 1,000 cells. After 2 hours (120 minutes), there have been 6 doublings: 1,000 × 2⁶ = 64,000. After 6 hours, 18 doublings: 1,000 × 2¹⁸ = 262,144,000. The growth looks slow at first and then explosive.
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For a growth rate r per period, the doubling time is ln 2 ÷ ln(1 + r). At 2% a year: 0.6931 ÷ 0.0198 = 35.0 years. At 1.5%: 46.6 years. At 3%: 23.4 years. The rule of 70 gives a quick estimate, 70 ÷ rate in percent: 70 ÷ 2 = 35. Similar logic applies to money, which is why the rule of 72 is used for investment returns.
A city of 500,000 growing at 2% a year will have 500,000 × 1.02¹⁰ = 609,500 people in 10 years, and 1,000,000 in about 35 years.
| Growth rate per year | Doubling time (exact) | Rule of 70 estimate |
|---|---|---|
| 1% | 69.7 years | 70 years |
| 1.5% | 46.6 years | 46.7 years |
| 2% | 35.0 years | 35 years |
| 3% | 23.4 years | 23.3 years |
| 5% | 14.2 years | 14 years |
| 7% | 10.2 years | 10 years |
Continuous Growth and the Rate Constant
When growth is continuous, N = N₀ × e^(kt), where k is the continuous rate. For bacteria with a 20-minute doubling time, k = ln 2 ÷ 20 = 0.0347 per minute, which corresponds to a growth of about 3.5% per minute compounding continuously. The two forms are equivalent: e^k = 1 + r per period. To fit data, plot the logarithm of the quantity against time; exponential growth appears as a straight line.
Limits to Growth
Real populations do not grow exponentially forever: food, space, and resources limit them. Bacterial growth levels off when nutrients run out, following an S-shaped logistic curve. Early epidemic growth is exponential, and the doubling time is a key measure: with a doubling time of 3 days, cases rise 1,024 times in 30 days if unchecked (2¹⁰). Exponential models are good for short periods when constraints are not binding, and poor for long-term forecasts.
Why It Is Counterintuitive
People underestimate exponential growth: a pond lily doubling daily that covers the pond on day 30 covers only half of it on day 29, and a quarter on day 28. A sheet of paper folded 42 times would reach the moon in theory. In finance, steady compounding turns modest annual returns into large sums over decades, and growing debts at high interest rates grow equally fast.
Frequently Asked Questions
What is the formula for exponential growth?
N = N₀ × (1 + r)^t.
How do I find the doubling time from a growth rate?
ln 2 ÷ ln(1 + r), or about 70 ÷ the percentage rate.
How long does a 2% annual growth rate take to double?
About 35 years.
How many bacteria after 6 hours with a 20-minute doubling time?
1,000 grows to about 262 million after 18 doublings.
Why can't exponential growth continue forever?
Because resources are limited.