I have to indicate for $f(n)=\log n$ and $g(n)=\sqrt[k]{n}$ if $f(n)=O(g(n))$ and if $g(n)=O(f(n))$.
For $f(n)=O(g(n))$:
I found it hard to prove it using the definition of big oh so I decided to use the limit approach. I have to take the limit of the division and determine if the limit exists:
$$ \lim_{n\to\infty}\frac{f(n)}{g(n)}\rightarrow\lim_{n\to\infty}\frac{\log n}{\sqrt[k]{n}} $$
If the limit exists then $f(n)=O(g(n))$. Is this correct?