Exponential growth and decay describe how a quantity changes over time at a rate proportional to its current value.
- Exponential growth happens when a quantity increases continuously by a fixed percentage over equal time intervals.
- Exponential decay happens when a quantity decreases continuously by a fixed percentage over equal time intervals.
These changes are commonly represented using the formulas:
where a is the initial value, r is the rate of change, and t is time.
Formulas of Exponential Growth and Decay
Exponential growth and decay can be represented using different but related formulas depending on the situation.
| Exponential Growth | Exponential Decay |
|---|---|
| f(x) = a·bˣ | f(x) = a·b⁻ˣ |
| f(x) = a(1 + r)ᵗ | f(x) = a(1 − r)ᵗ |
| P = P₀eᵏᵗ | P = P₀e⁻ᵏᵗ |
Here, a or P₀ is the initial value, b is the growth/decay factor, r is the rate, t or x represents time, k is a constant, and e ≈ 2.718. For growth, b = 1 + r = eᵏ, and for decay, b = 1 − r = e⁻ᵏ.
Applications of Exponential Growth and Decay
Various real-life processes in science, technology, and daily life follow the concept of exponential growth and decay.
- Bacterial Growth: Bacteria multiply rapidly (1, 2, 4, 8, …), demonstrating exponential growth in infections and diseases.
- Nuclear Reactions: Nuclear fusion represents exponential growth, while radioactive decay represents exponential decay.
- Feedback Spread: Information spreads quickly from one individual to many, especially through digital platforms.
- Computer Processing Power: Data storage and processing capabilities have grown exponentially (MB → GB → TB).
- Food Degradation: Food spoils slowly at first and then rapidly, illustrating exponential decay.
- Ageing Process: The decline in human health accelerates in later years, reflecting exponential decay.
- Internet Content Growth: Online content increases rapidly due to user activity and automated systems.
- Radioactive Decay: Substances like uranium decrease in quantity over time based on their half-life.
- Drug Elimination: The concentration of medicines in the body decreases gradually over time.
- Light Fading: Light intensity reduces exponentially as it passes through a medium.
- Car Depreciation: The value of a vehicle decreases significantly over time.
- Electrical Circuits: Electric charge in capacitors decreases exponentially.
- Population Decline: Species populations may decrease rapidly due to environmental factors.
- Learning and Memory: Retained information declines over time without regular revision.
- Financial Growth: Investments grow over time due to compound interest.