Prove the big O of this function $f(n) = n^4 + 10n^3\log(n)$ is $f(n) = O(n^4)$

Question

Use the formal definition of Big-O notation to prove his function $f(n) = n^4 + 10n^3\log(n)$ is $f(n) = O(n^4)$. I understand how to prove the Big O notation on polynomials mainly thanks to this post:

Big-O Notation - Prove that $n^2 + 2n + 3$ is $\mathcal O(n^2)$

but when there are logs in the question I am struggling.

Answer 1

For $n \ge 1,$ $\log(n) < n,$ so your function is bounded above by $11n^4,$ meaning it's "big oh" of $n^4.$