Big-O Notation regarding logarithms

Question

I got asked an interview question that wanted me to discern the Big-O notation of several logarithmic functions. The functions were as follows:

f(x) = log⁵

Answer 1

This is a matter of math:

1. f(x) = log⁵(x)
2. f(x) = log(x⁵) = 5 * log x
3. f(x) = log(6*log x) = log 6 + log(log x)
4. f(x) = log(log x)

So the Big O is

1. O(log⁵(x))
2. O(log x)
3. O(log (log x))
4. O(log (log x))

So (1) and (2) aren't equivalent, but (3) and (4) are (though they're different from both (1) and (2))