Is the following statement true or false?
$(\log n)^{10} = O(n^{0.10})$
When trying to solve this, I thought it was false, but according to my teacher's answers, it's true.
I would like to know if there is a quick and easy way to solve exercises like this?
Any power of logarithm grows slower than any positive power of $n$. If you look at the limit of $\frac{\log ((\log n)^k)}{\log ( n^p)} = \frac{ k \log \log n}{p \log n} \to 0$ since $\log \log n / \log n \to 0$ for any $k,p$. So, there exists a $N$ such tat for all $n \geq N$, $(\log n)^{10} \leq n^{0.10}$ (else the limit described prior would be a constant or tending to infinity).