Answer
$\left\{8\right\}$
Work Step by Step
Since $\log_b y=x$ implies $y=b^x$, the given equation, $
\log_4x=\dfrac{3}{2}
,$ implies
\begin{align*}\require{cancel}
4^{\frac{3}{2}}&=x
.\end{align*}
Using $b^{\frac{m}{n}}=\sqrt[n]{b^m}=\left(\sqrt[n]{b}\right)^m,$ the equation above is equivalent to
\begin{align*}\require{cancel}
\left(\sqrt{4}\right)^3&=x
\\
\left(2\right)^3&=x
\\
8&=x
\\
x&=8
.\end{align*}
Hence, the solution set of the equation $
\log_4x=\dfrac{3}{2}
$, is $
\left\{8\right\}
$.
