Question

Find the domains of definition of the following functions.
$y=\frac{logx}{arcsin(x-3)}$

Answer 1

$y=\frac{\log x}{{\sin }^{-1}(x-3)}$

${\sin }^{-1}(x-3)\ne 0⟹x-3\ne 0$

$⟹x\ne 3$

For $\log x$ to be defined, $x>0$

Hence the required solution is

$x\in (0,\infty )-\left\{3\right\}$