Exponential Growth and Decay
Exponential growth and decay describe quantities that increase or decrease by a fixed percentage over equal time intervals. Compound interest, population growth, radioactive decay, and viral spread all follow exponential models.
In this lesson
1 What Makes Growth Exponential
Linear growth adds the same amount every period. A savings account with a flat $100/month deposit grows linearly. Exponential growth multiplies by the same factor every period. An investment growing 8% per year grows exponentially, because 8% of a larger number is a larger absolute gain.
The key distinction: in linear growth, each period the same amount is added. In exponential growth, each period the same percentage is added , which means the absolute amount added grows each period because it's applied to a growing base.
People are bad at predicting exponential growth because we think linearly by default. A penny doubled every day for 30 days seems like it should stay small. Day 10: $5.12. Day 20: $5,242. Day 30: $5,368,709. The early days feel like nothing and then it explodes. This is why compound interest is so powerful and why epidemics are scary in their early stages. A penny doubled daily for 30 days: 1, 2, 4, 8... seems small at first. By day 30 it's $5.4 million. This counterintuitive acceleration is what makes exponential growth so powerful , and so dangerous when it's a disease or debt.
2 The Exponential Formula
A = P(1 + r)ᵗ for growth, A = P(1 − r)ᵗ for decay
A = final amount, P = initial amount (principal), r = rate per period (as a decimal), t = number of periods.
$8,954$13,3113 Doubling Time and Half-Life
Doubling time (for growth): the time it takes for a quantity to double. Rule of 72: divide 72 by the growth rate percentage to approximate doubling time. At 6% growth: 72/6 = 12 years to double.
Half-life (for decay): the time it takes for a quantity to halve. Used in radioactive decay, drug pharmacokinetics, and population decline.
4 Continuous Growth: e and Natural Log
When interest (or growth) compounds continuously rather than at discrete intervals: A = Peʳᵗ, where e ≈ 2.71828 and r is the continuous growth rate.
To find the time to reach a target: t = ln(A/P) / r. To find the rate given initial and final values over time t: r = ln(A/P) / t.
$2,7185 Real-World Applications
Compound interest: the foundational concept of personal finance. Money grows exponentially when returns compound. Time is the most powerful variable , starting early matters far more than the amount invested.
Epidemiology: early-stage disease spread is exponential. Each infected person infects R₀ others, who each infect R₀ more. When R₀ > 1, cases grow exponentially until interventions or immunity slow it. Understanding this is why early action in pandemics matters so much.
Physics: radioactive decay, Newton's law of cooling, and atmospheric pressure all follow exponential decay models. Moore's Law (transistor count doubling roughly every 2 years) described exponential growth in computing for decades.
Practice Problems
Sources & Further Reading
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