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To solve the expression \(\ln \left( \frac{2x \sqrt{1 + 4x}}{(x - 8)^4} \right)\), we need to simplify it by applying logarithmic properties. These properties include:

1. \(\ln \left( \frac{a}{b} \right) = \ln a - \ln b\) (logarithm of a quotient)
2. \(\ln (a \cdot b) = \ln a + \ln b\) (logarithm of a product)
3. \(\ln (a^n) = n \ln a\) (logarithm of a power)

### Step-by-step solution:

1. **Apply the quotient rule**:  
   \[
   \ln \left( \frac{2x \sqrt{1 + 4x}}{(x - 8)^4} \right) = \ln (2x \sqrt{1 + 4x}) - \ln ((x - 8)^4)
   \]

2. **Apply the power rule to the second term**:  
   Since \(\ln ((x - 8)^4) = 4 \ln (x - 8)\), we get:
   \[
   \ln (2x \sqrt{1 + 4x}) - 4 \ln (x - 8)
   \]

3. **Simplify the first term using the product rule**:
   \[
   \ln (2x \sqrt{1 + 4x}) = \ln (2x) + \ln (\sqrt{1 + 4x})
   \]

4. **Apply the power rule to the square root term**:
   Since \(\ln (\sqrt{1 + 4x}) = \frac{1}{2} \ln (1 + 4x)\), the expression becomes:
   \[
   \ln (2x) + \frac{1}{2} \ln (1 + 4x) - 4 \ln (x - 8)
   \]

5. **Simplify further using the product rule for \(\ln(2x)\)**:
   \[
   \ln(2x) = \ln 2 + \ln x
   \]

### Final simplified expression:
\[
\ln 2 + \ln x + \frac{1}{2} \ln (1 + 4x) - 4 \ln (x - 8)
\]

This is the simplified expression as a sum and difference of logarithms, with powers expressed as factors.

Would you like any further details or explanations? Here are some related questions to deepen your understanding:

1. How do you apply the quotient rule for logarithms in other cases?
2. What happens if the base of the logarithm is not \(e\) (natural log)?
3. Can you explain why \(\ln (\sqrt{a}) = \frac{1}{2} \ln(a)\)?
4. How would the expression change if the exponent on \((x - 8)\) were negative?
5. How can you further simplify the expression if \(x\) has specific values?

**Tip:** Always check if powers or square roots can be simplified early in logarithmic expressions.

Write the expression as a sum and/or difference of logarithms. Express powers as factors: ln(2x√(1 + 4x) / (x - 8)^4), x > 8.

Learn how to simplify the logarithmic expression ln(2x√(1 + 4x) / (x - 8)^4) using logarithmic properties like the quotient rule, power rule, and product rule. This problem demonstrates step-by-step how to break down the expression, making it suitable for students in Grades 10-12.

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