[Dylan16807]

That logic isn't right. You can implement an O(n log n) sort with whatever log base/radix you want. It could be 2, it could be 256, it could be 4096. If you use the same base/radix for both n and k, then log n is close to k and probably smaller.

Which version ends up faster is mostly based on factors that big O notation doesn't capture easily. You have to push things to impractical numbers like n=2^1000 or 2^1000000 to properly distinguish between constant factors and log factors, and at that point it doesn't actually help you write better code.