Answer
$\frac{1}{2}$ + $\frac{5}{2}$$\log_3 x$ - $\log_3 y$
Work Step by Step
$Expand$ $the$ $expression$:
$\log_3 \frac{\sqrt{3x^5}}{y}$
Apply the Second Law of Logarithms
$\log_3 \frac{\sqrt{3x^5}}{y}$ = $\log_3 \sqrt{3x^5}$ - $\log_3 y$
Rewrite the square root for 3x$^5$
$\log_3 (3x^5)^\frac{1}{2}$ - $\log_3 y$
Apply the Third Law of Logarithms for $\log_3 (3x^5)^\frac{1}{2}$
$\log_3 (3x^5)^\frac{1}{2}$ = $\frac{1}{2}$$\log_3 3x^5$
Apply the First Law of Logarithms for $\frac{1}{2}$$\log_3 3x^5$ (Also distribute the half as well)
$\frac{1}{2}$$\log_3 (3\times x^5)$ = $\frac{1}{2}$$\log_3 3$ + $\frac{1}{2}$$\log_3 x^5$
Use the Property $\log_x x$ $=$ $1$ for $\frac{1}{2}$$\log_3 3$
$\frac{1}{2}$$\log_3 3$ = $\frac{1}{2}$ $\times$ $1$ = $\frac{1}{2}$
Apply the Third Law of Logarithms for $\frac{1}{2}$$\log_3 x^5$
$\frac{1}{2}$$\log_3 x^5$ = $\frac{5}{2}$$\log_3 x$ [Note: $\frac{1}{2}$ $\times$ 5 = $\frac{5}{2}$]
Assemble the expression back together (Don't forget the $\log_3 y$)
$\frac{1}{2}$ + $\frac{5}{2}$$\log_3 x$ - $\log_3 y$