Q:

Is there any function with complexity O(log n) whereas power(2, f) is not in O(n)

Cant figure this one out:
Is there a function f such that: 
f(n) is in O(log n)

but, 
2^(f(n)) is not in O(n)

Should be correct, but I don't understand why.  

A:

Yes, consider the following example:
A = logn + logn/2, so A = O(log n) 
2A = 2logn + logn/2 =  2logn * 2logn/2 = n*n/2 = 1/2n2 ! = O(n)
(More specifically, 2A = ω(n), meaning it grows strictly faster than a linear function.)