Question

If $y=2^{\frac{1}{{\log }_{x}4}}$, then value of x in terms of y is

• A. 2y
• B. -2y
• C. $y^2$
• D. $-y^2$

Answer 1

The correct option is C $y^2$

$\mathrm{Since},y=2^{\frac{1}{{\log }_{x}4}}\mathrm{Taking}\log \mathrm{both}\mathrm{sides}\mathrm{with}\mathrm{base}2,\mathrm{we}\mathrm{get}\Rightarrow {\log }_{2}y=\frac{1}{{\log }_{x}4}\left(\mathrm{As}x>0,x\ne 1\right)\Rightarrow {\log }_{2}y={\log }_{4}x\left(\mathrm{As}{\log }_{a}b=\frac{1}{{\log }_{b}a}\right)\Rightarrow {\log }_{2}y={\log }_{2^2}x\Rightarrow {\log }_{2}y=\frac{1}{2}{\log }_{2}x\Rightarrow 2{\log }_{2}y={\log }_{2}x\Rightarrow x=y^2$