VC dimension of monotone disjunctions of length k over n variables?

Question

There are of course $n \choose k$ monotone disjunctions which bounds the VC dimension at $\log_2 {n \choose k}$. I'm wondering if this is bound at $k \log_2 n$? (Possibly follows from combinatorial identities).

More generally I'm looking at the claim in An algorithmic theory of learning: Robust concepts and random projection: "Further, it is NP-hard to learn a disjunction of $k$ variables as a disjunction of few than $k \log n$ variable." The author states this without proof; I'm assuming it's well-known or obvious but am trying to figure out why or where it is proved.