Question

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

• A. y = x
• B. y = x²
• C. y = x^1⁄2
• D. y = x³

Answer 1

The correct option is C

y = x^1⁄2

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

Taking log on both sides at base 2

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

${\log }_{2}y={\log }_{4}x$ $\left\{\begin{array}{cc}\because a^k=x& \\ {\log }_{a}x=k& \end{array}\right\}\left\{\begin{array}{cc}x\ne 1& \\ x>0& \end{array}\right\}$

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

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

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

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