nested log equation with a natural number root

Exercise:

Solve , ${{\log }_{4}}\left( 2{{\log }_{3}}\left( 1+{{\log }_{2}}\left( 1+3{{\log }_{3}}x \right) \right) \right)=\frac{1}{2}$

Solution: we know that ${{\log }_{a}}b=y\Leftrightarrow b={{a}^{y}}$

So $2{{\log }_{3}}\left( 1+{{\log }_{2}}\left( 1+3{{\log }_{3}}x \right) \right)={{4}^{1/2}}=2$

$\Rightarrow {{\log }_{3}}\left( 1+{{\log }_{2}}\left( 1+3{{\log }_{3}}x \right) \right)=1\Rightarrow 1+{{\log }_{2}}\left( 1+3{{\log }_{3}}x \right)={{3}^{1}}$

$\Rightarrow {{\log }_{2}}\left( 1+3{{\log }_{3}}x \right)=2\Rightarrow 1+3{{\log }_{3}}x={{2}^{2}}=4\Rightarrow 3{{\log }_{3}}x=3$

$\Rightarrow {{\log }_{3}}x=1={{\log }_{3}}3\Rightarrow x=3$