Answer
False
Work Step by Step
We are given the following equation:
$\log_3{\frac{3}{2}}\stackrel{?}{=}\frac{1}{2}\log_3{3}$
Recall the power property of logarithms (pg. 462):
$\log_b{m^n}=n\log_b{m}$
We apply this property to the right side of our expression:
$\log_3{\frac{3}{2}}\stackrel{?}{=}\log_3{3^{1/2}}$
$\log_3{\frac{3}{2}}\stackrel{?}{=}\log_3{\sqrt{3}}$
Note that $\log_a{m}=\log_a{n} \implies m=n$
Since $\sqrt3 \approx 1.73$, then
$\log_3{1.5}\ne\log_3{\sqrt3}$
Thus, we see that the expression is False.