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!