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The equation given is:

\[
x = 5^{\log_5(7) + \log_{\frac{1}{5}}(3)}
\]

Let's simplify this step by step.

1. **Simplify the first logarithm:**  
   \(\log_5(7)\) remains as it is.

2. **Simplify the second logarithm:**  
   \(\log_{\frac{1}{5}}(3)\) can be rewritten using the change of base formula:
   \[
   \log_{\frac{1}{5}}(3) = \frac{\log_5(3)}{\log_5\left(\frac{1}{5}\right)}
   \]
   We know that \(\log_5\left(\frac{1}{5}\right)\) simplifies to \(-1\) (since \(5^{-1} = \frac{1}{5}\)). Therefore:
   \[
   \log_{\frac{1}{5}}(3) = -\log_5(3)
   \]

3. **Combine the logarithms:**
   Now the expression becomes:
   \[
   x = 5^{\log_5(7) - \log_5(3)}
   \]

4. **Use the logarithmic property**:  
   The expression \(\log_5(7) - \log_5(3)\) can be simplified using the logarithm subtraction rule:
   \[
   \log_5\left(\frac{7}{3}\right)
   \]

   So, the expression simplifies to:
   \[
   x = 5^{\log_5\left(\frac{7}{3}\right)}
   \]

5. **Final simplification**:  
   Since \(5^{\log_5\left(\frac{7}{3}\right)}\) is simply \(\frac{7}{3}\), we have:
   \[
   x = \frac{7}{3}
   \]

So, the solution is:

\[
\boxed{\frac{7}{3}}
\]

Would you like more details or have any other questions?

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### Here are 5 related questions:

1. How do you simplify logarithms with different bases?
2. What are the properties of logarithms that allow you to combine them?
3. How do you convert between logarithms of different bases?
4. What happens when the base of the logarithm is a fraction?
5. How does the change of base formula work in logarithms?

### Tip:
Remember that logarithms are powerful tools for simplifying complex expressions, especially when they involve exponents. Understanding the fundamental properties of logarithms will help you manipulate and solve such equations effectively.

Solve for x in the equation x = 5^{\log_5(7) + \log_{\frac{1}{5}}(3)}.

Explore the solution to the equation x = 5^{\log_5(7) + \log_{\frac{1}{5}}(3)}. This problem involves logarithmic properties and the change of base formula, essential concepts in algebra. Suitable for students in Grades 9-12.

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