Exponential Growth and Decay Lesson

Algebra

What is exponential growth and decay?

Exponential growth and decay describe situations where a quantity increases or decreases by a fixed percentage over regular intervals of time.

Unlike arithmetic sequences (adding/subtracting a fixed amount), exponential changes multiply by a fixed factor.
These concepts appear in population growth, radioactive decay, interest rates, and more.

The general formula for exponential change is:

$\cdot b^t$

Where:

$b = 1 + r$

Where r = growth rate as a decimal

Example 1:

$b = 1 + 0.05 = 1.05$

Exponential growth formula:

$\cdot (1.05)^t$

$\cdot (1.05)^3 = 1000 \cdot 1.157625 \approx 1157.63$

✅ Population after 3 years ≈ 1158

$b = 1 - r$

Where r = decay rate as a decimal

Example 2:

$b = 1 - 0.10 = 0.9$

Exponential decay formula:

$\cdot (0.9)^t$

$\cdot (0.9)^5 = 200 \cdot 0.59049 \approx 118.10$

✅ Remaining substance ≈ 118 grams

Scenario	Formula	Growth or Decay
Population growth	y = 1500(1.02)^t	Growth, 2% per year
Bank interest (compounded annually)	y = 1000(1.05)^t	Growth, 5% per year
Radioactive decay	y = 500(0.8)^t	Decay, 20% per hour
Depreciating car value	y = 20000(0.85)^t	Decay, 15% per year

Sometimes you want to know how long it takes for a quantity to reach a certain value:

$\cdot b^t \quad \Rightarrow \quad t = \frac{\log(y/a)}{\log(b)}$

Example:

$\cdot 1.05^t$ $2 = 1.05^t$ $\frac{\log(2)}{\log(1.05)} \approx \frac{0.3010}{0.0212} \approx 14.2$

✅ About 14.2 years