Answer
$2\log_c x-\dfrac{3}{2}\log_c y-\log_cz$
Work Step by Step
Using the properties of logarithms, the given expression, $
\log_c \sqrt{\dfrac{x^4}{y^3z^2}}
,$ is equivalent to
\begin{array}{l}\require{cancel}
\log_c \left( \dfrac{x^4}{y^3z^2} \right)^{1/2}
\\\\=
\dfrac{1}{2}\log_c \dfrac{x^4}{y^3z^2}
\\\\=
\dfrac{1}{2}\left[ \log_c x^4-\log_c (y^3z^2) \right]
\\\\=
\dfrac{1}{2}\left[ \log_c x^4-\left( \log_c y^3+\log_cz^2 \right) \right]
\\\\=
\dfrac{1}{2}\left[ \log_c x^4-\log_c y^3-\log_cz^2 \right]
\\\\=
\dfrac{1}{2}\left[ 4\log_c x-3\log_c y-2\log_cz \right]
\\\\=
\dfrac{1}{2}(4\log_c x)-\dfrac{1}{2}(3\log_c y)-\dfrac{1}{2}(2\log_cz)
\\\\=
\dfrac{4}{2}\log_c x-\dfrac{3}{2}\log_c y-\dfrac{2}{2}\log_cz
\\\\=
2\log_c x-\dfrac{3}{2}\log_c y-\log_cz
.\end{array}
