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.