Answer
$\log_x{\left(\frac{2\sqrt{y}}{z^3}\right)}$
Work Step by Step
Recall the product property of logarithms (pg. 462):
$\log_b{mn}=\log_b{m}+\log_b{n}$
Applying this property, we get:
$\frac{1}{2}(\log_x{4}+\log_{x}{y})-3\log_x{z}=\frac{1}{2}\log_x{4y}-3\log_{x}{z}$
Next, recall the power property of logarithms (pg. 462):
$\log_b{m^n}=n\log_b{m}$
Applying this property, we get:
$\frac{1}{2}\log_x{4y}-3\log_{x}{z}\\
=\log_x{(4y)^{1/2}}-\log_x{z^3}\\
=\log_x{\sqrt{4y}}-\log_x{z^3}$
Finally, recall the quotient property of logarithms (pg. 462):
$\log_b{\frac{m}{n}}=\log_b{m}-\log_b{n}$
Applying this property, we get:
$\log_x{\sqrt{4y}}-\log_x{z^3}\\
=\log_x{\left(\frac{\sqrt{4y}}{z^3}\right)}\\
=\log_x{\left(\frac{2\sqrt{y}}{z^3}\right)}$
