To solve the equation \( x = 8 \log_8 13 \), let's proceed step by step.

1. **Understanding the equation**:  
   The expression involves the logarithm \(\log_8 13\), which means "the power to which 8 must be raised to get 13." The equation can be simplified as:

   \[
   x = 8 \times \log_8 13
   \]

2. **Simplifying the logarithm**:  
   Using the change of base formula, the logarithm \(\log_8 13\) can be expressed in terms of common logarithms (base 10) or natural logarithms (base \(e\)):

   \[
   \log_8 13 = \frac{\log_{10} 13}{\log_{10} 8}
   \]

   or

   \[
   \log_8 13 = \frac{\ln 13}{\ln 8}
   \]

3. **Substituting back into the equation**:  
   Now, substitute this back into the equation for \(x\):

   \[
   x = 8 \times \frac{\ln 13}{\ln 8}
   \]

4. **Calculating the value**:  
   We can calculate the logarithms and then multiply:

   \[
   \ln 13 \approx 2.5649
   \]
   \[
   \ln 8 \approx 2.0794
   \]

   So,

   \[
   x = 8 \times \frac{2.5649}{2.0794} \approx 8 \times 1.233 \approx 9.864
   \]

Thus, the value of \( x \) is approximately **9.864**.

Would you like more details on this calculation, or do you have any questions?

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Here are 8 related questions for further exploration:

1. What is the change of base formula for logarithms?
2. How would the equation change if \( \log_8 13 \) were replaced with \( \log_2 13 \)?
3. How do you convert logarithms of different bases?
4. What is the relationship between natural logarithms and common logarithms?
5. How do you interpret the value of \( \log_8 13 \) in terms of powers of 8?
6. How would the solution change if \( x = 2 \log_8 13 \) instead?
7. Can you calculate \( \log_8 13 \) using a calculator directly? How?
8. How does the base of the logarithm affect the value of the logarithm?

**Tip**: When dealing with logarithms, always remember the change of base formula, which allows you to convert any logarithm to a different base using natural or common logarithms.

Learn how to solve the equation x = 8 log8 13 step-by-step. Understand logarithms, change of base formula, and calculations involving logarithms of different bases. Suitable for advanced high school or college students.

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