I want to show that, for $n$ iid non-negative random variables $x_1, \dots, x_n$, we have $$ \mathbb{E} \max_{i \in [n]} x_i \lesssim (\mathbb{E} [x_i^{\log n}])^{\frac{1}{\log n }}. $$

I can only go so far as to show $\mathbb{E} \max_{i \in [n]} x_i \leq n (\mathbb{E} [x_i^{\log n}])^{\frac{1}{\log n }}$ for $n \geq 3$, using Jensen's inequality and the inequality $\max_i x_i \leq \sum_i x_i$, which seems a bit wasteful (and responsible for the factor of $n$).

Any tips for getting rid of the factor of $n$ would be greatly appreciated!