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If $f(x)$ and $g(x)$ are two functions such that

$$f(x) \not\in \Omega(g(x))$$

does

$$\log f(x) \not\in \Omega(\log g(x))$$

hold?

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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

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

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  • $\begingroup$ An even simpler example would be $f(x)=x$ and $g(x)=x^2$. $\endgroup$ – Discrete lizard Jan 19 '18 at 8:20
  • $\begingroup$ @Discretelizard Yeah. Thats a way better example. I was over thinking this one. $\endgroup$ – Jon Deaton Jan 19 '18 at 8:21

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