Question

If $x^{\left(\frac{1}{2}\right({\log }_{2}x{)}^2-\frac{1}{4}{\log }_{2}x-\frac{5}{4})}=\sqrt{2},$ then $x$ has

• A. only one integral value
• B. only one irrational value
• C. only one rational value
• D. two rational values

Answer 1

Answer: (A), (B), (D)

The correct options are
A only one integral value
B only one irrational value
D two rational values

$x^{\left(\frac{1}{2}\right({\log }_{2}x{)}^2-\frac{1}{4}{\log }_{2}x-\frac{5}{4})}=\sqrt{2}$
Taking $\log$ to base $2$ on both sides,
$\left[\frac{1}{2}\right({\log }_{2}x{)}^2-\frac{1}{4}{\log }_{2}x-\frac{5}{4}]{\log }_{2}x=\frac{1}{2}{\log }_{2}2$

Put ${\log }_{2}x=t,$ we get
$[\frac{1}{2}{t}^2-\frac{1}{4}t-\frac{5}{4}]t=\frac{1}{2}$
$\Rightarrow 2t^3-t^2-5t-2=0$
$\Rightarrow (t+1)(t-2)(2t+1)=0$
$\Rightarrow t=-1,2,-\frac{1}{2}$
$\Rightarrow {\log }_{2}x=-1,2,-\frac{1}{2}$
$\Rightarrow x=2^{-1},2^2,2^{-\frac{1}{2}}$
$\Rightarrow x=\frac{1}{2},4,\frac{1}{\sqrt{2}}$

One integral value: $4$
One irrational value: $\frac{1}{\sqrt{2}}$
Two rational values: $\frac{1}{2},4$