1. State the problem: Expand the logarithmic expression $$\log_3 \left[ \frac{18 (x + 2)^2}{(x - 2)^3 (x + 5)^2} \right]$$ using laws of logarithms. 2. Recall the logarithm laws useful here: - $$\log_b(AB) = \log_b A + \log_b B$$ - $$\log_b\left(\frac{A}{B}\right) = \log_b A - \log_b B$$ - $$\log_b(A^n) = n\log_b A$$ 3. Break down the expression inside the logarithm: - Numerator: $$18 (x + 2)^2$$ - Denominator: $$(x - 2)^3 (x + 5)^2$$ 4. Apply the quotient rule: $$\log_3 \left[ \frac{18 (x + 2)^2}{(x - 2)^3 (x + 5)^2} \right] = \log_3 \left(18 (x + 2)^2 \right) - \log_3 \left((x - 2)^3 (x + 5)^2\right)$$ 5. Apply the product rule to numerator and denominator: $$= \log_3 18 + \log_3 (x + 2)^2 - \left( \log_3 (x - 2)^3 + \log_3 (x + 5)^2\right)$$ 6. Apply the power rule to each term with exponents: $$= \log_3 18 + 2 \log_3 (x + 2) - \left( 3 \log_3 (x - 2) + 2 \log_3 (x + 5) \right)$$ 7. Distribute the minus sign to denominator terms: $$= \log_3 18 + 2 \log_3 (x + 2) - 3 \log_3 (x - 2) - 2 \log_3 (x + 5)$$ 8. The expression is now fully expanded as: $$\boxed{\log_3 18 + 2 \log_3 (x + 2) - 3 \log_3 (x - 2) - 2 \log_3 (x + 5)}$$ This is the expanded form of the given logarithmic expression using the laws of logarithms.