Asymptotic complexity

Question

If f(n) = O(g(n)),  is log(f(n)) = O(log(g(n))) ?

I just learnt today that this relation does not hold, because log changes the behavior of the functions. But is it true? An example will be good. Any help is appreciated.

Answer 1

Suppose f(n) =2pow(n) and g(n)=3pow(n)

Clearly f(n) = O(g(n))

but on taking log on both sides

log(f(n)) = nlog(2)

log(g(n)) = nlog(3)

 Now clearly log(f(n)) = Thetha(log(g(n)))

So the given statement is false