Use the properties of logarithms to expand the logarithm as much as possible. Rewrite the expression as a sum, difference, or product of logs. \[ \log \left(\sqrt{x^{5} y^{-6}}\right) \]

To expand the logarithm, we can use the properties of logarithms. Starting with your expression: \[ \log \left(\sqrt{x^{5} y^{-6}}\right) \] First, recall that \(\sqrt{a} = a^{1/2}\), so we can rewrite the expression: \[ \log \left((x^{5} y^{-6})^{1/2}\right) \] Next, we apply the power rule of logarithms, \(\log(a^b) = b \cdot \log(a)\): \[ \frac{1}{2} \log \left(x^{5} y^{-6}\right) \] Now we can further expand this using the product rule, \(\log(a \cdot b) = \log(a) + \log(b)\): \[ \frac{1}{2} \left( \log(x^{5}) + \log(y^{-6}) \right) \] Again, applying the power rule to each term: \[ \frac{1}{2} \left( 5 \log(x) + (-6) \log(y) \right) \] Which simplifies to: \[ \frac{5}{2} \log(x) - 3 \log(y) \] So the final expanded expression is: \[ \frac{5}{2} \log(x) - 3 \log(y) \]