Question

The number of integers satisfying the inequality ${\log }_2\sqrt{x}-2{\left({\log }_{4}x\right)}^2+1>0$ is/are

• A. $1$
• B. $2$
• C. $3$
• D. Infinitely many

Answer 1

The correct option is C $3$

${\log }_2\sqrt{x}-2({\log }_{4}x{)}^2+1>0\Rightarrow \frac{1}{2}{\log }_{2}x-\frac{2}{4}{\left({\log }_{2}x\right)}^2+1>0\Rightarrow {\left({\log }_{2}x\right)}^2-{\log }_{2}x-2<0\Rightarrow ({\log }_{2}x-2)({\log }_{2}x+1)<0\Rightarrow -1<{\log }_{2}x<2$
$\Rightarrow \frac{1}{2}<x<4$
The integers which will satisfy this equation are $x=1,2,3$