Answer
$\frac{1}{2} \log{x} + \frac{1}{2} \log{y}-3$
Work Step by Step
RECALL:
(1) $\log_b{(\frac{m}{n})}=\log_b{m} - \log_b{n}$
(2) $\log_b{(mn)}=\log_b{m} + \log_b{n}$
(3) $\log_b{(b^x)}=x$
(4) $\log_b{(m^n)}=n \cdot \log_b{m}$
(5) $\sqrt{a} = a^{\frac{1}{2}}$
Use rule (1) above to obtain:
$=\log{\sqrt{xy}}-\log{1000}$
Use rule (5) above to obtain:
$=\log{\left((xy){^\frac{1}{2}}\right)}-\log{1000}$
Write $1000$ as $10^3$ to obtain:
$=\log{\left((xy){^\frac{1}{2}}\right)}-\log{(10^3)}$
Use rule (3) above to obtain:
$=\log{\left((xy){^\frac{1}{2}}\right)}-3$
Use rule (4) above to obtain:
$=\frac{1}{2} \cdot \log{(xy)}-3$
Use rule (2) above to obtain:
$=\frac{1}{2}(\log{x} + \log{y})-3
\\=\frac{1}{2} \log{x} + \frac{1}{2} \log{y}-3$
