Question

Solve the inequality $lo{g}_{lo{g}_2\left(\frac{x}{2}\right)}(x^2-10x+22)>0$.

Answer 1

$log(x^2-10x+22)$

$lo{g}_2\left(\frac{x}{2}\right)>0$

$(x^2-10x+22)>lo{g}_2\left(\frac{x}{2}\right)$

$x^2-10x+22>1$

$x^2-10x+21>0$

$x^2-7x-3x+21>0$

$x(x-7)-3(x-7)>0$

$(x-3)(x-7)>0$

$\therefore (-\infty ,3)\cup (7,\infty )$

But $lo{g}_2^{\frac{x}{2}}$ should be defined

$lo{g}_2^{\frac{x}{2}}>0$

$\frac{x}{2}>1$

$x>2$

$\therefore (2,3)\cup (7,\infty )$.