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The question shown on the screen is as follows:

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**Problem:**
As a town gets smaller, the population of its high school decreases by 5% each year. The senior class has 320 students now. In how many years will it have about 100 students?

Write an equation and solve the equation.

In \( t = \) years, the senior class will have about 100 students.

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### Solution:

1. **Set up the equation:**
   Since the class size decreases by 5% each year, we can express the population in terms of exponential decay. The formula for exponential decay is:
   \[
   P = P_0 \times (1 - r)^t
   \]
   where:
   - \( P_0 \) is the initial population (320 students),
   - \( r \) is the decay rate (5%, or 0.05),
   - \( t \) is the number of years,
   - \( P \) is the population after \( t \) years.

   Substituting the values, we get:
   \[
   100 = 320 \times (1 - 0.05)^t
   \]
   Simplifying inside the parentheses:
   \[
   100 = 320 \times (0.95)^t
   \]

2. **Solve for \( t \):**
   Divide both sides by 320:
   \[
   \frac{100}{320} = (0.95)^t
   \]
   \[
   0.3125 = (0.95)^t
   \]
   
   Take the natural logarithm of both sides to solve for \( t \):
   \[
   \ln(0.3125) = t \cdot \ln(0.95)
   \]
   
   Now, divide by \( \ln(0.95) \):
   \[
   t = \frac{\ln(0.3125)}{\ln(0.95)}
   \]

3. **Calculate the value of \( t \):**
   Using approximate values:
   \[
   t \approx \frac{-1.16315}{-0.05129} \approx 22.68
   \]

### Answer:
It will take approximately **22.68 years** for the senior class to have about 100 students.

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### Further Questions
1. How does exponential decay differ from linear decay?
2. Can we solve this problem without logarithms? If so, how?
3. What would the equation look like if the population decreased by 10% each year instead?
4. How would the solution change if the target population was 150 students?
5. Why do we use logarithms in exponential decay equations?

### Tip
When dealing with exponential growth or decay, remember that logarithms are a helpful tool to solve for time when the exponent is the unknown variable.

As a town gets smaller, the population of its high school decreases by 5% each year. The senior class has 320 students now. In how many years will it have about 100 students? Write an equation and solve the equation.

Solve an exponential decay problem where a high school senior class's population decreases by 5% each year from 320 students. Find the number of years required for the population to reach about 100 students using exponential decay formulas and logarithms. Suitable for Grades 10-12.

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