Exponential Growth and Decay
Exponential growth occurs when a quantity increases by a constant percentage per time period — the absolute increase gets larger every period because it is always a fraction of a growing base. Decay is the reverse: decreasing by a constant percentage.
The General Formula
A = A₀ × e^(rt) (continuous)
A = A₀ × (1 + r)ᵗ (periodic)
A₀ = initial value
r = rate (positive = growth, negative = decay)
t = timeDoubling Time
t_double = ln(2) ÷ r ≈ 0.693 ÷ r
At 3% growth: doubles in 0.693/0.03 = 23.1 years
(Note: Rule of 72 gives 72÷3 = 24 — a good approximation)Half-Life (Decay)
t_half = ln(2) ÷ |r|
Carbon-14 decays at r ≈ 0.000121/year
Half-life = 0.693/0.000121 ≈ 5,730 yearsReal Applications
- Population growth, bacterial culture, viral spread
- Compound interest and investment growth
- Radioactive decay, drug clearance from the body
- Cooling (Newton's Law of Cooling)
- Epidemic modelling (exponential phase of spread)
Model exponential growth: Free Exponential Growth Calculator
The Exponential Growth Formula
A(t) = A₀ × eʳᵗ (continuous growth) or A(t) = A₀ × (1 + r)ᵗ (discrete growth), where A₀ = initial value, r = growth rate, t = time. The doubling time formula T₂ = ln(2)/r ≈ 0.693/r gives the time for any quantity to double. At 7% annual growth (r = 0.07), doubling time ≈ 0.693/0.07 ≈ 9.9 years. The Rule of 70 approximates this: divide 70 by the percentage growth rate to get doubling time in years.
Real-World Examples
- Population: World population grows at roughly 0.9%/year — doubling time ≈ 78 years.
- Investment: £1,000 at 8% per year doubles in 70/8 ≈ 8.75 years (Rule of 70).
- Bacteria: E. coli doubles every 20 minutes under ideal conditions — one cell becomes over 1 billion in 10 hours.
- Viral spread: An infection with R₀ = 3 (each case infects three others) grows exponentially until immunity or intervention slows it.
- Technology: Moore's Law — transistor counts on chips doubled roughly every 2 years for 50 years.
Frequently Asked Questions
What is the difference between exponential and linear growth?
Linear growth adds a constant amount per period: 10, 20, 30, 40... Exponential growth multiplies by a constant factor: 10, 20, 40, 80... Early on, exponential and linear look similar; over time, exponential growth dwarfs linear — and this non-intuitive gap causes people to consistently underestimate exponential processes.
What is exponential decay?
Exponential decay uses the same formula with a negative rate: A(t) = A₀ × e⁻ᵏᵗ. Radioactive decay, drug clearance from the body, and cooling of an object all follow exponential decay. The half-life (time to halve) = ln(2)/k ≈ 0.693/k. Carbon-14 has a half-life of 5,730 years, used in radiocarbon dating.
Why does compound interest grow faster than simple interest?
Simple interest: A = P(1 + rt). Compound interest: A = P(1 + r/n)^(nt). In compound interest, interest is added to principal and then earns interest itself. At 10% for 10 years: simple interest gives £2,000 on £1,000; annual compounding gives £1,000 × 1.1¹⁰ = £2,593.74 — 30% more. Continuous compounding (A = Peʳᵗ) gives £1,000 × e¹ = £2,718.28 — the mathematical maximum.