$(\tan 1 . \tan 89) . (\tan 2 . \tan 88) \ldots (\tan 44 . \tan 46) . \tan 45$
$= ( \tan 1 . \cot 1).(\tan 2 . \cot 2)\ldots (\tan 44. \cot 44). \tan 45$
$= 1 . 1 . 1 \ldots . 1 . \tan 45$
$= 1$
So, the given expression reduces to $\log 1 = 0.$
Correct Option: (C) 0