Question

Evaluate ${\log }_3({\log }_{9}x+\frac{1}{2}+9^x)=2x$

Answer 1

${\log }_3[{\log }_{9}x+\frac{1}{2}+9^x]=2x$

$\Rightarrow {\log }_{9}x+\frac{1}{2}+3^{2x}=3^{2x}$ ( using property of $\log$ )

$\Rightarrow {\log }_{9}x+\frac{1}{2}=0$

${\log }_{9}x=\frac{-1}{2}$

$x=9^{-1/2}$

$x=\frac{1}{9^{1/2}}$

$\therefore x=\frac{1}{3}$

Hence, the answer is $\frac{1}{3}.$