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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
  2. The exponential formula
  3. Doubling time and half-life
  4. Continuous growth: e and ln
  5. Real-world applications

1 What Makes Growth Exponential

Linear growth adds a fixed amount per period: a savings account gaining $100/month. Exponential growth multiplies by a fixed factor per period: an investment growing 8% per year.

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.

Why exponential growth surprises us

Humans intuitively think linearly. Exponential growth starts slowly then accelerates dramatically. 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.

Exponential Growth
$5,000 invested at 6% annual interest for 10 years.
1A = P(1 + r)ᵗ = 5000(1.06)¹⁰
21.06¹⁰ ≈ 1.7908
3A = 5000 × 1.7908 ≈ $8,954
Answer: $8,954 — nearly double the original investment
Exponential Decay
A car worth $30,000 depreciates 15% per year. Value after 5 years?
1A = P(1 − r)ᵗ = 30000(1 − 0.15)⁵ = 30000(0.85)⁵
20.85⁵ ≈ 0.4437
3A ≈ 30000 × 0.4437 ≈ $13,311
Answer: $13,311 — less than half the original value in 5 years

3 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.

Half-Life
Carbon-14 has a half-life of 5,730 years. Starting with 100g, how much remains after 17,190 years?
117,190 / 5,730 = 3 half-lives
2After 1 half-life: 50g. After 2: 25g. After 3: 12.5g
Answer: 12.5g remains — this is how carbon dating works

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.

Continuous Compounding
$1,000 at 5% continuous compounding for 20 years.
1A = Peʳᵗ = 1000 × e^(0.05 × 20) = 1000 × e¹
2e¹ ≈ 2.71828
3A ≈ $2,718
Answer: $2,718 — compared to $2,653 with annual compounding

5 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

$2,000 invested at 4% annually for 8 years. Final value?
A = 2000(1.04)⁸ = 2000 × 1.3686 ≈ $2,737
Using the Rule of 72, how long does it take $10,000 to double at 9% annual growth?
72 / 9 = 8 years
A population of 500 bacteria grows at 20% per hour. How many after 6 hours?
A = 500(1.20)⁶ = 500 × 2.986 ≈ 1,493 bacteria