For which value of $u$ will $2^{u}>u^{100}$ start to hold?
This task can be traced back to problem 217. Let $u=2^{x}$, then the inequality

$$
2^{2^{x}}>2^{100 x}
$$

must hold. Since the larger of two powers of 2 is the one with the larger exponent, the inequality is equivalent to $2^{x}>100 x$. This inequality holds for $x \geq 10$, and thus the inequality holds for $u \geq 2^{10}=1024$.