Introduction
Exponential and logarithmic functions are used to model real-world situations such as population growth, radioactive decay, interest, and sound intensity. Understanding these applications allows us to solve practical problems.
Examples
Example 1: Population Growth
A town has 5,000 people and grows at 3% per year. Find the population after 10 years.
Solution: P = 5000(1 + 0.03)^10 ≈ 5000(1.3439) ≈ 6719
Example 2: Radioactive Decay
A radioactive substance has 200 grams and decays at 8% per year. Find the remaining amount after 5 years.
Solution: A = 200(1 − 0.08)^5 = 200(0.92)^5 ≈ 200(0.659) ≈ 131.8 grams
Example 3: Continuous Growth
A bank account has $1,000 and earns 5% annual interest compounded continuously. Find the balance after 3 years.
Solution: A = 1000·e^(0.05·3) = 1000·e^0.15 ≈ 1000·1.1618 ≈ $1161.82
Example 4: Richter Scale
The magnitude M of an earthquake is calculated as M = log(I/Iâ‚€), where I is the intensity and Iâ‚€ is a reference intensity. If I/Iâ‚€ = 10,000, find M.
Solution: M = log(10,000) = 4
Practice Problems
- A population of 2,000 grows by 4% per year. Find the population in 8 years.
- A substance decays at 12% per year. If you start with 500 grams, how much remains after 3 years?
- An investment of $2,000 grows at 6% annually, compounded continuously. Find the balance after 5 years.
- The intensity of an earthquake is 50,000 times the reference intensity. Find its Richter magnitude.
- A bacteria culture doubles every 3 hours. If you start with 150 bacteria, how many are present after 12 hours?