I think this is the way your teacher expects you to find the
answer:

log338 means 

the exponent to which 3 must be raised to give 38.
So to find the least integer greater than that exponent, 
we begin by trying integer exponents of 3 until we find 
the smallest integer such that when 3 is raised to that 
power, the answer is greater that 38.  

We begin by trying exponent 1:

31 = 3  <--smaller than 38, so we try the next integer 2.

32 = 3×3 = 9  <--still smaller than 38, so we try 3

33 = 3×3×3 = 27  <--still smaller than 38, so we try 4

34 = 3×3×3×3 = 81  <--that's it!  for 81 is greater than 38

So the least integer greater than log338 is 4.

Edwin