Question

The number of integral value(s) of $x$ that satisfy the inequality $({\log }_{0.5}x{)}^2+{\log }_{0.5}x-2\le 0$ is

Answer 1

$({\log }_{0.5}x{)}^2+{\log }_{0.5}x-2\le 0$
$\log$ function is defined $x>0$
$\because {\log }_{0.5}x=-{\log }_{2}x$
Let ${\log }_{2}x=z$
$\Rightarrow z^2-z-2\le 0$
$\Rightarrow (z-2)(z+1)\le 0$
$\Rightarrow z\in [-1,2]$
$\therefore {\log }_{2}x\in [-1,2]$
$\Rightarrow x\in [\frac{1}{2},4]$
$\therefore x=1,2,3,4$