Answer
$x=\dfrac{3}{4}$
Work Step by Step
Changing to exponential form, the given equation, $
\log_x \dfrac{9}{16}=2
,$ is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{9}{16}=x^2
.\end{array}
Taking the square root of both sides, the solutions of the equation above are
\begin{array}{l}\require{cancel}
x=\pm\sqrt{\dfrac{9}{16}}
\\\\
x=\pm\dfrac{3}{4}
.\end{array}
Upon checking, only $
x=\dfrac{3}{4}
$ satisfies the original equation.