How to find the Big-O of $\log(\binom{n}{\log n})$

The definition of Big-O is $$u_n = O(v_n) \iff (\exists c \in \mathbb{R}^{*})\,\, (\exists N \in \mathbb{N}) \,\, n > N \implies u_n < c \, v_n$$

Quest

Based on that I am trying to find the upper bound for this function $$\log(\binom{n}{\log n})$$, given that $$\binom{n}{k} \le \frac{n^k}{k!} \le (\frac{n\cdot e}{k})^k$$

What I've done so far

What I've done so far is $$\log(\binom{n}{\log n}) \le \log\left(\left(\frac{n\cdot e}{\log n}\right)^{\log n}\right)$$ but it doesn't seem to lead anywhere.

Any ideas?

• Use Stirling's fomrula – Mike Hawk Apr 2 at 14:42