Use the properties of logarithms to condense the following expression as much as possible, writing the answer as a single term with a coefficient of 1 . All exponents should be positive. \[ 5\left(\ln \left(\sqrt[3]{e^{2}}\right)-\ln (x y)\right) \]

To condense the expression, we first apply the properties of logarithms. Start with the inside of the parentheses: 1. \(\ln \left(\sqrt[3]{e^{2}}\right)\) can be rewritten using the power rule: \[ \sqrt[3]{e^{2}} = e^{\frac{2}{3}} \implies \ln \left(\sqrt[3]{e^{2}}\right) = \frac{2}{3} \ln(e) = \frac{2}{3} \] since \(\ln(e) = 1\). 2. Next, we rewrite the expression inside the parentheses: \[ \ln \left(\sqrt[3]{e^{2}}\right) - \ln (x y) = \frac{2}{3} - \ln(xy) \] 3. Using the quotient rule for logarithms, we have: \[ \frac{2}{3} - \ln(xy) = \ln\left(\frac{e^{\frac{2}{3}}}{xy}\right) \] 4. Now, multiply the entire expression by 5: \[ 5\left(\ln\left(\frac{e^{\frac{2}{3}}}{xy}\right)\right) = \ln\left(\left(\frac{e^{\frac{2}{3}}}{xy}\right)^{5}\right) \] 5. Simplifying this further: \[ \left(\frac{e^{\frac{2}{3}}}{xy}\right)^{5} = \frac{e^{\frac{10}{3}}}{(xy)^{5}} \] 6. Finally, we can condense the entire expression: \[ \ln\left(\frac{e^{\frac{10}{3}}}{x^5y^5}\right) \] Thus, the final result is: \[ \ln\left(\frac{e^{\frac{10}{3}}}{x^5y^5}\right) \]