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To solve the given exponential equation \( e^{5x} = 8 \) and express the solution set in terms of natural logarithms, follow these steps:

1. **Take the natural logarithm of both sides**:  
   \( \ln(e^{5x}) = \ln(8) \).

2. **Simplify using logarithm properties**:  
   The logarithm of an exponential expression simplifies as:  
   \( 5x = \ln(8) \).

3. **Solve for \( x \)**:  
   Divide both sides by 5 to isolate \( x \):  
   \[
   x = \frac{\ln(8)}{5}.
   \]

Thus, the solution is \( x = \frac{\ln(8)}{5} \).

### Answer:
The correct answer from the given options is \( \frac{\ln(8)}{5} \), which corresponds to the first option: \( \frac{\ln(8)}{5} \).

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Would you like more details or have any further questions?

Here are five related questions to expand on this topic:

1. How do logarithm properties simplify the equation-solving process?
2. What is the general method for solving exponential equations like this?
3. How does changing the base of the exponential function affect the solution?
4. Can you solve \( e^{3x} = 5 \) using a similar approach?
5. What is the interpretation of natural logarithms in real-world contexts?

**Tip**: When solving equations involving exponents, logarithms are your most useful tool for "undoing" the exponential function.

Solve the exponential equation. Express the solution set in terms of natural logarithms. e^{5x} = 8

Learn how to solve the exponential equation e^{5x} = 8 and express the solution in terms of natural logarithms. This detailed explanation covers key logarithmic properties and algebraic manipulation. Suitable for high school students in Grades 9-12.

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