Answer
$\dfrac {1}{2}\log _{3}x+\dfrac {1}{3}\log _{3}y-2\log _{3}w-\dfrac {1}{2}\log _{3}z$
Work Step by Step
$\log _{3}\dfrac {\sqrt {x}\times \sqrt [3] {y}}{w^{2}\sqrt {z}}=\log _{3}(x^{\frac {1}{2}}\times y^{\frac {1}{3}}\times w^{-2}\times z^{-\frac {1}{2}})=\dfrac {1}{2}\log _{3}x+\dfrac {1}{3}\log _{3}y-2\log _{3}w-\dfrac {1}{2}\log _{3}z$