Necessary conditions for proving If f(n) = O(g(n)), then is log(f(n)) = O(log(g(n))) I am learning about algorithmic complexities and I read that  if f(n) and g(n) are asymptotically positive functions and if  $f(n) =O(g(n))$ then the relationship $log(f(n)) = O(log(g(n)))$ holds.

I would like to know what are the necessary and sufficient conditions for this relation to hold?
It's obviously wrong. As an example take f(n) = 2, g(n) = $1 + e^{-n}$. 

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