Question 12 - of 21 step 1 of 1 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. \[ 5\left(\ln \left(\sqrt[3]{e^{7}}\right)-\ln (x y)\right) \]

Given the expression \[ 5\left(\ln \left(\sqrt[3]{e^{7}}\right)-\ln (xy)\right), \] we proceed with the following steps: 1. Rewrite the radical as an exponent: \[ \sqrt[3]{e^7} = e^{\frac{7}{3}}. \] 2. Replace the logarithm of the exponential: \[ \ln\left(e^{\frac{7}{3}}\right) = \frac{7}{3}. \] 3. Substituting into the expression: \[ 5\left(\frac{7}{3} - \ln(xy)\right). \] 4. Distribute the coefficient 5: \[ \frac{35}{3} - 5\ln(xy). \] 5. Use the power property of logarithms, writing \(5 \ln(xy)= \ln\left((xy)^5\right)\): \[ \frac{35}{3} - \ln\left((xy)^5\right). \] 6. Express the constant \(\frac{35}{3}\) as a logarithm. Recall that \(\frac{35}{3} = \ln\left(e^{\frac{35}{3}}\right)\): \[ \ln\left(e^{\frac{35}{3}}\right) - \ln\left((xy)^5\right). \] 7. Use the quotient property of logarithms to combine into one logarithm: \[ \ln\left(\frac{e^{\frac{35}{3}}}{(xy)^5}\right). \] Thus, the condensed expression is \[ \ln\left(\frac{e^{\frac{35}{3}}}{(xy)^5}\right). \]