multiple of an integer and asymptotics

Question

Let us suppose that we have a positive integer $N$.

We take the integer $\lceil \log_2 N \rceil$. Does there always exist an integer $X \geq N$ such that the following both conditions are satisfied:

1) $X$ is a multiple of $\lceil \log_2 N \rceil$

2) $X=O(N)$

?

Answer 1

Hint. One may take $$ X= \lceil N/\log_2 N \rceil \lceil \log_2 N \rceil. $$