Exponential Growth & Decay
Exponential growth and decay describe situations where quantities multiply by a constant factor over equal time periods. The general form y = a · bˣ captures this pattern, where a represents the initial amount, b is the growth factor, and x is the number of time periods. When b > 1, the quantity grows exponentially; when 0 < b < 1, it decays exponentially.
Why it matters
Exponential functions model critical real-world phenomena across multiple fields. In finance, compound interest follows exponential growth — $1,000 invested at 8% annually becomes $2,159 after 10 years. Population biology relies on exponential models to predict species growth or decline. Radioactive decay in nuclear physics follows exponential patterns, with carbon-14 having a half-life of 5,730 years for archaeological dating. Medical professionals use exponential decay to calculate drug concentrations in bloodstreams. Technology adoption often follows exponential curves — smartphone users grew from 122 million in 2007 to 6.8 billion by 2023. Understanding exponential functions prepares students for advanced topics including logarithms, calculus derivatives, and differential equations. Climate science uses exponential models for atmospheric CO₂ growth, while epidemiologists apply them to disease spread patterns.
How to solve exponential growth & decay
Exponential Growth
- General form: y = a · bˣ, where a is the starting value and b is the growth factor.
- If b > 1, the quantity grows; if 0 < b < 1, it decays.
- Percent growth of r% means b = 1 + r/100.
- To find y after x periods, substitute and evaluate.
Example: A population of 500 grows 10% per year. After 3 years: y = 500 · 1.10³ ≈ 665.5.
Worked examples
A bacteria colony starts with 50 bacteria and doubles every hour. How many bacteria are there after 2 hours?
Answer: 200
- Identify the doubling pattern → 50 × 2² — The colony doubles 2 times, so multiply by 2².
- Calculate the power → 2² = 4 — 2 multiplied by itself 2 times is 4.
- Multiply by the starting amount → 50 × 4 = 200 — There are 200 bacteria after 2 hours.
A town has 5,000 people and grows by 25% per year. How many people live there after 2 years?
Answer: 7812
- Find the growth factor → 1 + 25100 = 1.25 — A 25% increase means multiplying by 1.25 each year.
- Year 1 → 5000 × 1.25 = 6250 — After year 1 the population is 6250.
- Year 2 → 6250 × 1.25 = 7812 — After year 2 the population is 7812.
- Verify with formula → A = 5000 × 1.25² = 7812 — Using A = P × (1 + r)ᵗ confirms the answer.
A car worth $300,000.00 loses 10% of its value each year. What is it worth after 4 years?
Answer: $196,830.00
- Find the decay factor → 1 − 10100 = 0.9 — Losing 10% means multiplying by 0.9 each year.
- Year 1 → 300000 × 0.9 = 270000 — After year 1 the value is $270,000.00.
- Year 2 → 270000 × 0.9 = 243000 — After year 2 the value is $243,000.00.
- Year 3 → 243000 × 0.9 = 218700 — After year 3 the value is $218,700.00.
- Year 4 → 218700 × 0.9 = 196830 — After year 4 the value is $196,830.00.
- Verify with formula → A = 300,000 × 0.9⁴ = 196,830 — Using A = P × (1 − r)ᵗ confirms the answer.
Common mistakes
- Confusing linear and exponential growth by adding the same amount each period instead of multiplying by the growth factor, such as calculating 100 bacteria after 3 hours of doubling as 100 + 50 + 50 + 50 = 250 instead of 100 × 2³ = 800.
- Using the percentage rate directly as the growth factor, such as treating 15% growth as multiplying by 0.15 instead of 1.15, leading to 1,000 × 0.15³ = 3.375 instead of the correct 1,000 × 1.15³ = 1,520.875.
- Applying the exponent to the initial value instead of the growth factor, calculating 500 bacteria doubling for 4 hours as 500⁴ × 2 = 125,000,002 instead of 500 × 2⁴ = 8,000.