# How do you expand ln(sqrt(((xy^2)/z))?

May 17, 2016

For this problem, we will use the following $\ln$ properties:

• ln^a = aln
• ln(x/y) = lnx - lny
• ln(xy) = lnx + lny

First, let's establish that $\sqrt{a} = {a}^{\frac{1}{2}}$

Therefore, $\ln \left(\sqrt{\frac{x {y}^{2}}{z}}\right) = \ln {\left(\frac{x {y}^{2}}{z}\right)}^{\frac{1}{2}}$

$= \frac{1}{2} \ln \left(\frac{x {y}^{2}}{z}\right)$

$= \frac{1}{2} \left(\ln x {y}^{2} - \ln z\right)$

$= \frac{1}{2} \left(\ln x + \ln {y}^{2} - \ln z\right)$

Hopefully this helps!