Proposition 11

Let $g$ be a function in $\mathscr{H}(\mathfrak{F}, \mathbf{R})$ such that $\lim_{\mathfrak{F}} g = +\infty$; the relation $f \ll g$ implies $e^f \ll e^g$; the relation $f \sim g$ implies $\log f \sim \log g$.

Indeed, if $f \ll g$ then $f - g = g \left( \frac{f}{g} - 1 \right)$ tends to $-\infty$ along $\mathfrak{F}$. Similarly, if $f \sim g$ one has $\log f = \log g + \log \frac{f}{g}$, so $\log f - \log g$ tends to 0, and the same for $\frac{\log f}{\log g} - 1 = \frac{\log f - \log g}{\log g}$.

On the other hand, note that the relation $f \sim g$ does not imply $e^f \sim e^g$, nor even that $e^f \asymp e^g$, as is shown by the example where $f(x) = x^2, \ g(x) = x^3 + x$ as $x$ tends to $+\infty$; similarly, the relation $f \ll g$ does not imply $\log f \ll \log g$, as is shown by the example $f(x) = x, \ g(x) = x^2$ as $x$ tends to $+\infty$.