To expand the given logarithmic expression using the laws of logarithms, let's first rewrite the expression:

$\log \left( \frac{x^2 + 8}{(x^2 + 7)(x^3 - 4)^2} \right)$

Step 1: Apply the Quotient Rule of Logarithms

The Quotient Rule states:

$\log \left( \frac{A}{B} \right) = \log(A) - \log(B)$

Here, $A = x^2 + 8$ and $B = (x^2 + 7)(x^3 - 4)^2$.

Applying this rule:

$\log \left( \frac{x^2 + 8}{(x^2 + 7)(x^3 - 4)^2} \right) = \log(x^2 + 8) - \log((x^2 + 7)(x^3 - 4)^2)$

Step 2: Apply the Product Rule of Logarithms

The Product Rule states:

$\log(AB) = \log(A) + \log(B)$

In this case, we need to apply the product rule to the second term:

$\log((x^2 + 7)(x^3 - 4)^2) = \log(x^2 + 7) + \log((x^3 - 4)^2)$

Thus, we have:

$\log(x^2 + 8) - \left( \log(x^2 + 7) + \log((x^3 - 4)^2) \right)$

Step 3: Apply the Power Rule of Logarithms

The Power Rule states:

$\log(A^n) = n\log(A)$

Apply the power rule to the term $\log((x^3 - 4)^2)$:

$\log((x^3 - 4)^2) = 2\log(x^3 - 4)$

Final Expanded Expression

Substituting this into the expression, we get the final expanded form:

$\log(x^2 + 8) - \log(x^2 + 7) - 2\log(x^3 - 4)$

This is the expanded form of the given logarithmic expression.

---

Do you want any further details or clarifications?

Here are 5 related questions you might find useful:

1. How would you apply logarithmic rules to simplify more complex expressions?
2. What happens if you have a base other than 10 or $e$ in a logarithmic expansion?
3. How can the Laws of Logarithms be applied to solve logarithmic equations?
4. What are common mistakes to avoid when expanding logarithmic expressions?
5. How do logarithmic properties help in solving real-world problems like exponential growth?

Tip: Always apply the quotient rule before the product rule when both are present in an expression!