Asymptotics of logarithms of functions If I know that $\lim\limits_{x\to \infty} \dfrac{f(x)}{g(x)}=1$, does it follow that $\lim\limits_{x\to\infty} \dfrac{\log f(x)}{\log g(x)}=1$ as well? I see that this definitely doesn't hold for $\dfrac{e^{f(x)}}{e^{g(x)}}$ (take $f(x)=x+1$ and $g(x)=x$), but I'm not sure how to handle the other direction.

It does not follow. Take the example of $f(x)=e^{-x}+1$ and $g(x)=1$. Then 
$$
\lim_{x \to \infty}\frac{f(x)}{g(x)}= \lim_{x \to \infty} \frac{e^{-x}+1}{1}=1
$$
However, 
$$
\lim_{x \to \infty} \frac{\log f(x)}{\log g(x)}=\lim_{x \to \infty} \frac{\log(e^{-x}+1)}{\log 1}
$$
Does not exist.
