Answer
$\log_p \dfrac{x^3y^{1/2}}{z^{3/2}a^3}$
Work Step by Step
Using the properties of logarithms, the given expression, $
3\log_p x+\dfrac{1}{2}\log_p y-\dfrac{3}{2}\log_p z-3\log_p a
$, is equivalent to
\begin{align*}
&
\log_p x^3+\log_p y^{\frac{1}{2}}-\log_p z^{\frac{3}{2}}-\log_p a^3
&(\text{use }\log_b x^y=y\log_b x)
\\\\&=
\left(\log_p x^3+\log_p y^{\frac{1}{2}}\right)-\left(\log_p z^{\frac{3}{2}}+\log_p a^3\right)
\\\\&=
\log_p \left(x^3y^{\frac{1}{2}}\right)-\log_p \left(z^{\frac{3}{2}}a^3\right)
&(\text{use }\log_b (xy)=\log_b x+\log_b y)
\\\\&=
\log_p \dfrac{x^3y^{\frac{1}{2}}}{z^{\frac{3}{2}}a^3}
&(\text{use }\log_b \dfrac{x}{y}=\log_b x-\log_b y)
\\\\&=
\log_p \dfrac{x^3y^{1/2}}{z^{3/2}a^3}
.\end{align*}
Hence, the expression $
3\log_p x+\dfrac{1}{2}\log_p y-\dfrac{3}{2}\log_p z-3\log_p a
$ is equivalent to $
\log_p \dfrac{x^3y^{1/2}}{z^{3/2}a^3}
$.
