If function f and g given by $f(x) = \log (x-1) - \log (x-2)$ and $g(x) = \log(\frac{x-1}{x-2})$ then x lies in the interval

(2, ∞)

(x) is defined for all x satisfying

$x-1>0$ and $x-2>0$ i.e. $x >2$

∴Domain $(f)=(2, ∞)$    ...(i)

g(x) is defined for all x satisfying

$\frac{x-1}{x-2}>0⇒x∈ (-∞, 1) ∪ (2,∞)$

Domain $(g)=(-∞, 1) ∪(2, ∞) $   ...(ii)

Thus, $f(x)$ and $g (x)$ are equal for all x belonging to their common domain i.e. (2, ∞).