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

(2, ∞)

The correct answer is Option (3) → (2, ∞)

f(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, ∞).