Limit of log with factor

Question

Problem

If $$\lim_{n \rightarrow \infty} f(n) - \log n - \log \log n = y$$ where $f$ is some function, does this imply that $$ \lim_{n \rightarrow \infty} f(n) - \log (xn) - \log \log n = y$$ for some $x \in O(1)$? Or does this give another Limit $y$?

Progress

For $y=0$ or $y= \pm \infty$, I think, this factor $x$ should not change anything, but for $y\in (-\infty,0) \cup (0,\infty)$, I think this Limit should change, shouldn't it?

Answer 1

Notice that

$$\log(xn)=\log x+\log n$$ so the limit is the same if $y=\pm\infty$ and it becomes $y+\log x$ if $y\in\Bbb R$.

Answer 2

Hint:

$\log (ab) = \log a + \log b$