#### Answer

Domain: $[2, 10)$

#### Work Step by Step

We need to find the domain of the given function:
$h(x)=\sqrt{x-2}-\log_{5}(10-x)$
We know that we can't take the log of a negative number (or 0), so:
$10-x\gt 0$
$10\gt x$
Also, we can't square root a negative:
$x-2\geq 0$
$x\geq 2$
Thus overall: $2\leq x\lt 10$
Therefore, the domain is: $[2, 10)$.