Answer
$x=\{ 1, 10 \}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
\log x=\sqrt{\log x}
,$ square both sides of the equation. Then use concepts of solving quadratic equations to solve for the values of $x$. Finally, do checking of the solutions with the original equation.
$\bf{\text{Solution Details:}}$
Squaring both sides of the equation above results to
\begin{array}{l}\require{cancel}
\left( \log x \right)^2=\left( \sqrt{\log x} \right)^2
\\\\
\left( \log x \right)^2=\log x
.\end{array}
Transposing $\log x$ and factoring the $GCF$ of the resulting expression result to
\begin{array}{l}\require{cancel}
\left( \log x \right)^2-\log x=0
\\\\
\log x(\log x-1)=0
.\end{array}
Equating each factor to $0$ (Zero Product Property), the solutions are
\begin{array}{l}\require{cancel}
\log x=0
\\\\\text{OR}\\\\
\log x-1=0
.\end{array}
Since $\log_by=x$ is equivalent to $y=b^x$, the equations above, in exponential form, are equivalent to
\begin{array}{l}\require{cancel}
\log x=0
\\\\
\log_{10} x=0
\\\\
x=10^0
\\\\
x=1
\\\\\text{OR}\\\\
\log x-1=0
\\\\
\log_{10} x=1
\\\\
x=10^1
\\\\
x=10
.\end{array}
Upon checking, both $
x=\{ 1, 10 \}
,$ satisfy the original equation.