Answer
$\log_2{\left(\dfrac{(x-y)^{\frac{3}{2}}}{(x^2+y^2)^2}\right)}$
Work Step by Step
RECALL:
(1) $n \cdot \log_a{P}=\log_a{(P^n)}$
(2) $\log_a{P} + \log_a{Q}=\log_a{(PQ)}$
(3) $\log_a{P} - \log_a{Q}=\log_a{(\frac{P}{Q})}$
(4) $a^m \cdot a^n=a^{m+n}$
Use rule (1) above to obtain
$\frac{3}{2}\log_2{(x-y)}-2\log{(x^2+y^2)}= \log_2{[(x-y)^{\frac{3}{2}}]}-\log_2{[(x^2+y^2)^2])}$
Use rule (3) above to obtain:
$\log_2{[(x-y)^{\frac{3}{2}}]}-\log_2{[(x^2+y^2)^2])}=\log_2{\left(\dfrac{(x-y)^{\frac{3}{2}}}{(x^2+y^2)^2}\right)}$
