Answer
-0.2263
Work Step by Step
From the change of base formula, we know that $log_{b}a=\frac{log_{c}a}{log_{c}b}$ ($a$, $b$, and $c$ are positive real numbers and neither $b$ nor $c$ is equal to 1).
Therefore, $log_{6}\frac{2}{3}=\frac{log_{10}\frac{2}{3}}{log_{10}6}=\frac{-0.1761}{0.7782}=-0.2263$.