Logarithm Expansion 0Cecc1

1. **State the problem:** Expand the logarithmic expression $$\ln\left(\frac{x^8}{\sqrt{x^2 - 36}}\right)$$ 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:** $$\ln\left(\frac{x^8}{\sqrt{x^2 - 36}}\right) = \ln(x^8) - \ln\left(\sqrt{x^2 - 36}\right)$$ 4. **Rewrite the square root as an exponent:** $$\ln(x^8) - \ln\left((x^2 - 36)^{\frac{1}{2}}\right)$$ 5. **Apply the power rule:** $$8 \ln(x) - \frac{1}{2} \ln(x^2 - 36)$$ 6. **Final expanded form:** $$\boxed{8 \ln(x) - \frac{1}{2} \ln(x^2 - 36)}$$ This is the expanded logarithmic expression using the laws of logarithms.