Answer
$\displaystyle \ln 3+\frac{1}{2}\ln 5-\frac{1}{3}\ln 6$
Work Step by Step
...Apply rule: $\displaystyle \log_{a}\frac{x}{y}=\log_{a}x-\log_{a}y$
$\displaystyle \ln\frac{3\sqrt{5}}{\sqrt{3}{6}}=$
$=\ln(3\cdot\sqrt{5})-\ln(\sqrt[3]{6})$
... Apply rule: $\log_{a}xy=\log_{a}x+\log_{a}y$
$=\ln 3+\ln\sqrt{5} \ln(\sqrt[3]{6})$
... $\sqrt[3]{6}=6^{1/3}, \sqrt{5}=5^{1/2}$
$=\ln 3+\ln 5^{1/2}-\ln 6^{1/3}$
... Apply rule: $ \log_{a}x^{r}=r\log_{a}x$
$=\displaystyle \ln 3+\frac{1}{2}\ln 5-\frac{1}{3}\ln 6$
