Suppose that $f (x) =O(g(x))$. Does it follow that $\log |f (x)| =O(log |g(x)|)$?

Question

Suppose that $f(x)=O(g(x))$. Does it follow that $\log |f (x)|=O(log |g(x)|)$? I start from $f(x)=O(g(x))$, until I get

Does it mean $\log |f (x)|=O(log |g(x)|)$?

Answer 1

No;

If $f \equiv 5$, $g \equiv 1$, then $f = O(g)$, but $\log g = 0, \log f > 0$.

The desired result is true, for example, in the case that $g(x) \rightarrow \infty$ as $x \rightarrow \infty, |g(x)| > A > 1 \;\forall x \in \mathbb{R}$: If $C>0$ is an admissible constant such that $|f(x)|\le C|g(x)|$, then $\log|f(x)| \le \log C + \log|g(x)| \le 2 \log |g(x)| $ for sufficiently large $x$. The small cases will yield a different constant, but in the end you're going to have the desired result.