The given expression is:

\[
\ln\left(\frac{x^6}{x^2 - 25}\right), \quad x > 5
\]

### Steps to Simplify:

1. **Use the properties of logarithms**: The logarithmic property \(\ln \left( \frac{a}{b} \right) = \ln(a) - \ln(b)\) allows us to break the expression into two parts:

   \[
   \ln \left( \frac{x^6}{x^2 - 25} \right) = \ln(x^6) - \ln(x^2 - 25)
   \]

2. **Simplify further**: Using another logarithmic property \(\ln(a^n) = n \ln(a)\), the first part becomes:

   \[
   \ln(x^6) = 6 \ln(x)
   \]

   So the expression becomes:

   \[
   6 \ln(x) - \ln(x^2 - 25)
   \]

Thus, the simplified expression is:

\[
6 \ln(x) - \ln(x^2 - 25), \quad x > 5
\]

Would you like further details on this, or have any questions?

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Here are 5 related questions:

1. How do logarithmic properties help in simplifying complex expressions?
2. Can you explain the domain restriction \(x > 5\) for this function?
3. What would happen if we differentiate the simplified expression?
4. How do you expand or factor \(x^2 - 25\) using algebraic methods?
5. What is the significance of \(x^6\) in the original logarithmic equation?

**Tip:** When simplifying logarithmic expressions, always check for opportunities to apply properties like product, quotient, and power rules!

Simplify the logarithmic expression ln(x^6 / (x^2 - 25)) with a step-by-step explanation using logarithmic properties. This problem involves algebraic concepts and is suitable for students in Grades 10-12.

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