## Summary of Exponential Growth and Decay

### Essential Concepts

• Exponential growth and exponential decay are two of the most common applications of exponential functions.
• Systems that exhibit exponential growth follow a model of the form $y={y}_{0}{e}^{kt}.$
• In exponential growth, the rate of growth is proportional to the quantity present. In other words, ${y}^{\prime }=ky.$
• Systems that exhibit exponential growth have a constant doubling time, which is given by $(\text{ln}2)\text{/}k.$
• Systems that exhibit exponential decay follow a model of the form $y={y}_{0}{e}^{\text{−}kt}.$
• Systems that exhibit exponential decay have a constant half-life, which is given by $(\text{ln}2)\text{/}k.$

## Glossary

doubling time
if a quantity grows exponentially, the doubling time is the amount of time it takes the quantity to double, and is given by $\frac{(\text{ln}2)}{k}$
exponential decay
systems that exhibit exponential decay follow a model of the form $y={y}_{0}{e}^{\text{−}kt}$
exponential growth
systems that exhibit exponential growth follow a model of the form $y={y}_{0}{e}^{kt}$
half-life
if a quantity decays exponentially, the half-life is the amount of time it takes the quantity to be reduced by half. It is given by $\frac{(\text{ln}2)}{k}$