Answer
Their growth rates are comparable.
Work Step by Step
We will find the limit $\lim_{x\to\infty}\frac{\ln x^{20}}{\ln x}.$
1) If it is equal to zero then $\ln x$ grows slower than $\ln x$;
2) If it is equal to $\infty$ then $\ln x^{20}$ grows faster than $\ln x$;
3) If it is equal to some ňon zero constant then their growth rates are comparable.
We will use the logarithmic rule $\ln b^a=a\ln b$:
$$\lim_{x\to\infty}\frac{\ln x^{20}}{\ln x}=\lim_{x\to\infty}\frac{20\ln x}{\ln x}=\lim_{x\to\infty}=20,$$
thus 3) is right and their growth rates are comparable.
