Question

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

• A. $y=x$
• B. $y=x^2$
• C. $y=x^{\frac{1}{2}}$
• D. $y=x^3$

Answer 1

The correct option is C

$y=x^{\frac{1}{2}}$

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

Taking log on both sides with base 2

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

${\log }_{2}y={\log }_{2^2}x$ $\left\{\begin{array}{cc}\because a^k=x& \\ {\log }_{a}x=k& \end{array}\right\}\left\{\begin{array}{cc}x\ne 1& \\ x>0& \end{array}\right\}$

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

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

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

$y=x^{\frac{1}{2}}$