$${{log}}_{{\mathtt{4}}}{\left({\mathtt{x}}\right)} = {\sqrt{{{log}}_{{\mathtt{4}}}{\left({\mathtt{x}}\right)}}}$$
$$log_4(4) = 1$$
$$\\\log_4(x)=\sqrt{\log_4(x)} \\
\\
\quad x=4 \Rightarrow \log_4(4)=\sqrt{\log_4(4)}\Rightarrow 1 = \sqrt1 = 1$$
$$\boxed{x=4}$$
...or
$$\\\log_4(x)=\sqrt{\log_4(x)} \quad | \quad 1^2\\ \\
\log_4(x)\times\log_4(x)=\log_4(x) \quad | \quad :\log_4(x)\\ \\
\log_4(x) = 1\quad | \quad 4^x\\ \\
4^{\log_4(x)} =x= 4^1 \\
\boxed{x=4}$$