Necessary conditions for proving If f(n) = O(g(n)), then is log(f(n)) = O(log(g(n)))
It's obviously wrong. As an example take f(n) = 2, g(n) = $1 + e^{-n}$. 

log f(n) = 1, log g(n) ≈ $e^{-n} / \log e$.