Question

If ${\left(\frac{x}{2}\right)}^{{\log }_{2}x}=8x,$ then the values of $x$ is/are

• A. $\frac{1}{4}$
• B. $\frac{1}{2}$
• C. $4$
• D. $8$

Answer 1

Answer: (B), (D)

The correct options are
B $\frac{1}{2}$
D $8$

${\left(\frac{x}{2}\right)}^{{\log }_{2}x}=8x$
Taking ${\log }_2$ both sides, we get
$\Rightarrow {\log }_2\left({\left(\frac{x}{2}\right)}^{{\log }_{2}x}\right)={\log }_{2}8x$
$\Rightarrow {\log }_{2}x{\log }_2\left(\frac{x}{2}\right)={\log }_{2}8+{\log }_{2}x$
$\Rightarrow {\log }_{2}x({\log }_{2}x-1)=3+{\log }_{2}x$
Taking ${\log }_{2}x=y$
$y(y-1)=3+y\Rightarrow y^2-2y-3=0\Rightarrow (y-3)(y+1)=0\Rightarrow y=-1,3\Rightarrow {\log }_{2}y=-1,3\therefore y=\frac{1}{2},8$