Question

If $y=2^{\frac{1}{{\log }_{x}4}}$, then

Answer 1

$y=2^{\frac{1}{{\log }_{x}4}}$

Taking log on both sides with base $2$

$\Rightarrow {\log }_{2}y={\log }_2{2}^{\frac{1}{{\log }_{x}4}}$
$\left\{\begin{array}{cc}\because a^k=x& ;x\ne 1\\ {\log }_{a}x=k& ;x>0\end{array}\right\}$
$\Rightarrow {\log }_{2}y=\frac{1}{{\log }_{x}4}$
$[{\log }_{a}b=\frac{1}{{\log }_{b}a}]$
$\Rightarrow {\log }_{2}y={\log }_{4}x$

$\Rightarrow {\log }_{2}y={\log }_{2^2}x$

$\Rightarrow {\log }_{2}y=\frac{1}{2}{\log }_{2}x$
$[{\log }_{a^k}x=\frac{1}{k}{\log }_{a}x]$

$\Rightarrow {\log }_{2}y={\log }_2{x}^{\frac{1}{2}}$

$\Rightarrow y=x^{\frac{1}{2}}$
Hence the correct answer is Option C.