Question

Sum of integers satisfying $\sqrt{{\log }_{2}x-1}-\frac{1}{2}{\log }_2\left(x^3\right)+2>0$ is

Answer 1

$\sqrt{{\log }_{2}x-1}-\frac{1}{2}{\log }_2\left(x^3\right)+2>0x>0\text{and}{\log }_{2}x-1\ge 0\Rightarrow x>0\text{and}x\ge 2$

$\Rightarrow \sqrt{{\log }_{2}x-1}-\frac{3}{2}{\log }_{2}x+2>0$
$\Rightarrow \sqrt{{\log }_{2}x-1}-\frac{3}{2}({\log }_{2}x-1)+\frac{1}{2}>0$
Let
$\sqrt{{\log }_{2}x-1}=t\ge 0$
$\Rightarrow t-\frac{3}{2}{t}^2+\frac{1}{2}>0$
$\Rightarrow 3t^2-2t-1<0\Rightarrow (3t+1)(t-1)<0$
$\Rightarrow -\frac{1}{3}<t<1$
So,
$0\le t<1$

$\Rightarrow 0\le \sqrt{{\log }_{2}x-1}<1$
$\Rightarrow 0\le {\log }_{2}x-1<1$
$\Rightarrow 1\le {\log }_{2}x<2$
$\Rightarrow 2\le x<4$
Hence, the integral values are $2,3$
Sum $=5$