Growth rate of roots vs logarithms
Using L'Hôpital's rule:
$$
\lim_{n \to \infty} \frac{n^{1/10}}{\log n} = 
\lim_{n \to \infty} \frac{\frac{1}{10} n^{-9/10}}{n^{-1}} =
\lim_{n \to \infty} \frac{n^{1/10}}{10} = +\infty.
$$