Question

If $a,b$ are the maximum and minimum integral values of $x$ which satisfy the inequality $\frac{{\log }_{4}x-2}{{\log }_{4}x-5}<0$ respectively, then the value of $a-b$ is

Answer 1

$\frac{{\log }_{4}x-2}{{\log }_{4}x-5}<0$
Let ${\log }_{4}x=y$
$\Rightarrow \frac{y-2}{y-5}<0\Rightarrow y\in (2,5)\therefore 2<{\log }_{4}x<5\Rightarrow 16<x<1024\Rightarrow x\in (16,1024)$
So,
$a=1023,b=17\therefore a-b=1023-17=1006$