Asymptotic complexity class preservation under logarithm?
This does not hold for all $f$, $g$. Consider $f(x) = x!$ and $g(x) = x^x$. It is the case that $x! \not\in \Omega(x^x)$ yet, using Stirling's approximation, we have that

$$\log (x!) \in \Theta(x \log x)$$

Since, $\log(x^x) = x \log x$, we have

$$\log f(x) \in \Theta(\log g(x))$$