Mathematics July 22, 2026 · 11 min read

Exponential Growth: Why Things Speed Up Over Time

Calculate exponential growth and decay. Find doubling time, growth factor, and final values with worked examples.

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Exponential growth is when a quantity increases by a constant percentage per time period. Unlike linear growth (which adds a fixed amount), exponential growth multiplies by a fixed factor, causing the quantity to accelerate dramatically over time.

The Formula

P(t) = P₀ × (1 + r)^t

Where P₀ is the initial value, r is the growth rate, and t is time.

Doubling Time

T_d = ln(2)/ln(1 + r) ≈ 70/r% (Rule of 70)

Worked Example

1000 grows at 3% for 20 years:

  1. P(20) = 1000 × (1.03)^20 ≈ 1806.1
  2. Doubling time ≈ 70/3 ≈ 23.3 years

Applications

  • Finance: Compound interest, investment growth
  • Population: Bacteria, city populations
  • Technology: Moore\'s Law, computing power
  • Epidemiology: Early-stage disease spread