Answer
$$\frac{1}{2} \log x + \frac{1}{2}\log y -3$$
Work Step by Step
We simplify as follows:
$$\log \left ( \frac{\sqrt{xy}}{1000} \right )= \log \sqrt{xy} - \log 1000= \log (xy)^{\frac{1}{2}}-\log 10^3= \frac{1}{2} \log xy -3 \log 10= \frac{1}{2} \log x + \frac{1}{2}\log y -3$$
