Use the Laws of Logarithms to expand the expression.

log 

*x*2 + 8

(*x*2 + 7)(*x*3 − 4)2

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.

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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!

Use the Laws of Logarithms to expand the expression log (x^2 + 8)/((x^2 + 7)(x^3 − 4)^2).

Learn how to expand a complex logarithmic expression using the quotient, product, and power rules. This step-by-step guide explains how to apply the Laws of Logarithms to expressions involving algebraic terms and exponents, ideal for students in Grades 10-12.

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