Question

The number of integral solutions satisfying the equation $\sqrt{{\log }_{2}x-1}-\frac{1}{2}{\log }_2\left(x^3\right)+2>0$ is equal to

Answer 1

$\sqrt{{\log }_{2}x-1}-\frac{1}{2}{\log }_2\left(x^3\right)+2>0$
$\sqrt{{\log }_{2}x-1}-\frac{3}{2}{\log }_2\left(x\right)+2>0[\because \log x^m=m\log x]$
Let ${\log }_{2}x$ be $t^2+1.$
$\therefore$ the question looks like $t-\frac{3}{2}(t^2+1)+2>0$
$\Rightarrow 2t-3t^2-3+4>0$
$\Rightarrow 3t^2-2t-1<0$
$\Rightarrow 3t^2-3t+t-1<0$
$\Rightarrow (t-1)(3t+1)<0$
So, $t\in (\frac{-1}{3},1)$
Hence, ${\log }_{2}x\in (\frac{10}{9},2)$
So, $x\in (2^{\frac{10}{9}},4)\in (2^{1.11111},4)\in (2.16,4)$
In this range, only one integer exists, i.e. $3$