1. **State the problem:** Expand the logarithmic expression $$\ln \left( \frac{x^3}{\sqrt{x^2 - 36}} \right)$$ using the laws of logarithms, given that $$x > 6$$. 2. **Recall the logarithm laws:** - $$\ln \left( \frac{a}{b} \right) = \ln a - \ln b$$ - $$\ln (a^n) = n \ln a$$ - $$\sqrt{c} = c^{\frac{1}{2}}$$ 3. **Apply the quotient rule:** $$\ln \left( \frac{x^3}{\sqrt{x^2 - 36}} \right) = \ln (x^3) - \ln \left( \sqrt{x^2 - 36} \right)$$ 4. **Rewrite the square root as an exponent:** $$\ln (x^3) - \ln \left( (x^2 - 36)^{\frac{1}{2}} \right)$$ 5. **Apply the power rule:** $$3 \ln x - \frac{1}{2} \ln (x^2 - 36)$$ 6. **Final expanded form:** $$\boxed{3 \ln x - \frac{1}{2} \ln (x^2 - 36)}$$ This is the expanded form of the given logarithmic expression using the laws of logarithms.