You don't need to change base to solve this problem.
log 4 to base 4 = 1 and log 4 to base 16 is 1/2. 4^1 =4 and 16^1/2 =4
So x = 3/2
The question is testing your understanding of logs,rather than just being able to use them without really knowing the theory.
The base 16 log of 4 means what' power do I need to raise 16 to in order to get 4.' and since sqrt 16 =4 ,it's 1/2.
Remember that a logarithm is an index,that always helps if you are getting confused between base and index.