对数与平方根的 Big-O

一般来说，以下内容总是正确的吗？

log(n) = O(n^a/(a+1))? s.t. a is any constant positive integer, perhaps very large.

如果不是的话，最大的值是多少a这个陈述对于哪些人来说是正确的？

As functions go, log(n) always grows slower than n^{any power} when n gets large. See https://math.stackexchange.com/questions/1663818/does-the-logarithm-function-grow-slower-than-any-polynomial https://math.stackexchange.com/questions/1663818/does-the-logarithm-function-grow-slower-than-any-polynomial for proof.

So as long as a is a constant positive integer, it really doesn't matter what its value is, as it will always be possible to find constants C and k such that log(n) <= |C(n^a/(a+1)| + k, which is the definition of big-O.

However, you can also understand it intuitively: n^a/(a+1) approaches n as a becomes large. Naturally, log(n) is always O(n).