To solve the equation \( 15 = 800 \left( 0.5 \right)^{\frac{x}{5}} \), follow these steps:

1. **Isolate the exponential expression**:
   \[
   \left( 0.5 \right)^{\frac{x}{5}} = \frac{15}{800}
   \]

2. **Simplify the fraction**:
   \[
   \frac{15}{800} = 0.01875
   \]
   Therefore, the equation becomes:
   \[
   \left( 0.5 \right)^{\frac{x}{5}} = 0.01875
   \]

3. **Apply the natural logarithm to both sides** to solve for \( x \):
   \[
   \ln \left( 0.5^{\frac{x}{5}} \right) = \ln(0.01875)
   \]

4. **Use the power rule of logarithms** \( \ln(a^b) = b \ln(a) \):
   \[
   \frac{x}{5} \ln(0.5) = \ln(0.01875)
   \]

5. **Solve for \( x \)**:
   \[
   x = 5 \frac{\ln(0.01875)}{\ln(0.5)}
   \]

6. **Calculate the logarithms**:
   \[
   \ln(0.01875) \approx -3.981
   \]
   \[
   \ln(0.5) \approx -0.693
   \]

7. **Substitute and solve**:
   \[
   x = 5 \frac{-3.981}{-0.693} \approx 5 \cdot 5.744 = 28.72
   \]

So, the value of \( x \) is approximately \( 28.72 \).

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How would you solve \( 20 = 500 (0.5)^{\frac{x}{4}} \)?
2. What is the effect of changing the base of the exponential function?
3. How can you solve \( 15 = 800 \left( 0.3 \right)^{\frac{x}{5}} \) instead?
4. What are the properties of logarithms that are useful in solving exponential equations?
5. How can you verify the solution of an exponential equation?

**Tip:** Always check your final solution by substituting it back into the original equation to ensure it satisfies the equation.

Learn how to solve the exponential equation 15 = 800 * (0.5)^(x/5) with detailed steps and calculations. This problem involves exponential functions, logarithms, and solving equations. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation