Question

Find the domain of definition of the following function:
$y=\sqrt{\frac{{\log }_{0.9}|x-2|}{|x|}}.$

Answer 1

$\frac{{\log }_{0.9}|x-2|}{|x|}\ge 0\Rightarrow x\ne 0$

As base of $\log$ is less than $1$

$|x-2|\le 10<|x-2|\le 1$

$\pm (x-2)\le 1x-2\le 1x\ne 2x\le 3$ and $x-2\ge -1x\ge 1$

Taking intersection, we get

$x\in [1,3]-\left\{2\right\}$