The Cauchy condensation test:
$$\sum_{n=2}^\infty\frac1{(\log(n))^{\log(n)}}<\sum_{n=1}^\infty\frac{2^n}{(\log(2^n))^{\log(2^n)}}=\sum_{n=1}^\infty\frac{2^n}{n^{n\log(2)}(\log(2))^{n\log(2)}}<\sum_{n=1}^\infty\frac{2^n}{n^n}$$
And that last sum converges by ratio test, hence your series converges.