Question

Let $y=2^{\frac{1}{{\log }_{x}4}}$. If $x=9$, then the value of $y$ is

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

The correct option is A $3$

Since $y=2^{\frac{1}{{\log }_{x}4}}$,
Taking log both sides on base 2
${\log }_{2}y=\frac{1}{{\log }_{x}4}$ $(\because x>0,x\ne 1)$
or ${\log }_{2}y={\log }_{4}x[\because {\log }_{a}b=\frac{1}{{\log }_{b}a}]=\frac{1}{2}{\log }_{2}x[\because {\log }_{a^m}b=\frac{1}{m}{\log }_{a}b]$
or $2{\log }_{2}y={\log }_{2}x$ or ${\log }_2{y}^2[\because m\log a=\log a^m]={\log }_{2}x$
$\therefore x=y^2$
$\therefore x=9,y=3$
Ans: A