k log n is the number of bits it takes to represent a random sequence of k elements from S. So f(S) is the number of bits one can remember when they're encoded as sequences of elements of S.

My guess is that f(S) will be smaller when n is very small (it's not twice as easy to remember 20 digits from {0,1} as to remember 10 from {0,1,2,3}, unless perhaps you explicitly convert them) and maybe when n is rather large (as it starts to require cognitive effort to distinguish elements of S). It will surely also depend in complicated ways on what sort of thing is in S (words? common first names? photos of faces? ...).

I'm sure there's been work on this in the context of computer security -- looking for things that are a bit like passwords but make a better memorability/bits-of-security tradeoff than passwords do. In that context it's also worth considering what happens if you allow the sequence to be remembered slightly wrongly.