$\begin{align}  & \frac{\log \sqrt{27}+\log \sqrt{8}-\log \sqrt{125}}{\log 5-\log 6}=\frac{\log {{3}^{\tfrac{3}{2}}}+\log {{2}^{\tfrac{3}{2}}}-\log {{5}^{\tfrac{3}{2}}}}{\log 5-\log 6} \\ & \frac{\log \sqrt{27}+\log \sqrt{8}-\log \sqrt{125}}{\log 5-\log 6}=\frac{\tfrac{3}{2}\log 3+\tfrac{3}{2}\log 2-\tfrac{3}{2}\log 5}{\log 5-\log 6} \\ & \frac{\log \sqrt{27}+\log \sqrt{8}-\log \sqrt{125}}{\log 5-\log 6}=\frac{\tfrac{3}{2}(\log 3+\log 2-\log 5)}{\log 5-\log 6} \\ & \frac{\log \sqrt{27}+\log \sqrt{8}-\log \sqrt{125}}{\log 5-\log 6}=\frac{\tfrac{3}{2}(\log 6-\log 5)}{\log 5-\log 6} \\ & \frac{\log \sqrt{27}+\log \sqrt{8}-\log \sqrt{125}}{\log 5-\log 6}=-\frac{3}{2} \\\end{align}$