Table of Contents
Exponential Growth
Exponential growth occurs when a quantity increases at a rate proportional to its current value. The larger it gets, the faster it grows. This creates the characteristic J-shaped curve seen in population growth, compound interest, viral spread, and technology adoption.
Exponential decay is the reverse: radioactive decay, drug metabolism, and depreciation follow negative exponential rates. Both are modeled by the same equation with positive (growth) or negative (decay) rate parameters.
Formulas
Growth Examples
| Scenario | Rate | Doubling Time |
|---|---|---|
| Bacteria (optimal) | ~69%/hr | 1 hour |
| World population | ~1.1%/yr | 63 years |
| Compound interest 7% | 7%/yr | ~10 years |
| Moore's Law | ~41%/yr | ~2 years |
Rule of 72
A quick mental math shortcut: divide 72 by the growth rate percentage to estimate doubling time. At 6% growth, doubling takes about 72/6 = 12 periods.
Frequently Asked Questions
Continuous vs discrete growth?
Continuous uses e^(rt), discrete uses (1+r)^t. For small rates they are nearly identical. Continuous is used in physics and biology; discrete in finance (annual compounding).
Can growth be sustained forever?
No. Real-world exponential growth eventually hits resource limits, transitioning to logistic (S-shaped) growth. Pure exponential models are accurate only for early/unconstrained growth phases.
How to find growth rate from data?
r = ln(y_final / y_initial) / t. Given initial and final values with elapsed time, this formula extracts the continuous growth rate.