# How do you use the properties of logarithms to expand lnroot3(x/y)?

Jul 4, 2018

$\ln \sqrt[3]{\frac{x}{y}} = \frac{1}{3} \ln x - \frac{1}{3} \ln y$
$\ln \sqrt[3]{\frac{x}{y}}$
$= \ln {\left(\frac{x}{y}\right)}^{\frac{1}{3}}$
$= \frac{1}{3} \times \ln \left(\frac{x}{y}\right)$
$= \frac{1}{3} \times \left(\ln x - \ln y\right)$
$= \frac{1}{3} \ln x - \frac{1}{3} \ln y$