Answer
$\log_a \dfrac{\sqrt{a}}{x}$
Work Step by Step
Using the properties of logarithms, the given expression, $
\log_a \dfrac{a}{\sqrt{x}}-\log_a \sqrt{ax}
,$ is equivalent to
\begin{array}{l}\require{cancel}
\log_a \dfrac{\dfrac{a}{\sqrt{x}}}{\sqrt{ax}}
\\\\=
\log_a \dfrac{a}{\sqrt{x}\cdot\sqrt{ax}}
\\\\=
\log_a \dfrac{a}{\sqrt{ax^2}}
\\\\=
\log_a \dfrac{a}{x\sqrt{a}}
\\\\=
\log_a \left( \dfrac{a}{x\sqrt{a}}\cdot\dfrac{\sqrt{a}}{\sqrt{a}} \right)
\\\\=
\log_a \dfrac{a\sqrt{a}}{x\cdot a}
\\\\=
\log_a \dfrac{\sqrt{a}}{x}
.\end{array}
