Answer
$\frac{1}{2}\log{2}+\frac{1}{2}\log{x}-\frac{1}{2}\log{y}$
Work Step by Step
Recall the basic properties of logarithms (pg. 462):
Product Property:
$\log_b{mn}=\log_b{m}+\log_b{n}$
Quotient Property:
$\log_b{\frac{m}{n}}=\log_b{m}-\log_b{n}$
Power Property:
$\log_b{m^n}=n\log_b{m}$
We are given:
$\log{\sqrt{\frac{2x}{y}}}$
First, we apply the power property:
$=\log{(\frac{2x}{y})^{1/2}}$
$=\frac{1}{2}\log{(\frac{2x}{y})}$
Next, we apply the quotient property:
$=\frac{1}{2}(\log{2x}-\log{y})$
$=\frac{1}{2}\log{2x}-\frac{1}{2}\log{y}$
We could expand further with the product property, if we wish:
$=\frac{1}{2}\log{2}+\frac{1}{2}\log{x}-\frac{1}{2}\log{y}$
