# How do you expand ln (sqrt(ex^2)/y^3)?

Jul 6, 2016

$\frac{1}{2} + \ln x - 3 \ln y$

#### Explanation:

Expanding this expression is done by applying two properties of $\ln$
Quotient property:
$\ln \left(\frac{a}{b}\right) = \ln a - \ln b$

Product property:
$\ln \left(a \cdot b\right) = \ln a + \ln b$

$L n \left(\frac{\sqrt{e {x}^{2}}}{y} ^ 3\right)$
$= \ln \left(\sqrt{e {x}^{2}}\right) - \ln \left({y}^{3}\right)$
$= \ln \left({\left(e {x}^{2}\right)}^{\frac{1}{2}}\right) - 3 \ln y$
$= \frac{1}{2} \ln \left(e {x}^{2}\right) - 3 \ln y$
$= \frac{1}{2} \left(\ln e + \ln \left({x}^{2}\right)\right) - 3 \ln y$
$= \frac{1}{2} \left(1 + 2 \ln x\right) - 3 \ln y$
$= \frac{1}{2} + \ln x - 3 \ln y$