Asymptotic solutions for inequalities

Question

How do I determine the order (big o) of $\omega$ in $e^{-\omega/\epsilon}\leq10^{-9}$ and $e^{-\omega/\epsilon}\leq\epsilon$, where $\epsilon$ is a small parameter.

Answer 1

In the first case, taking logs yields $$-\omega/\epsilon \leq -9 \ln(10)\\ \omega \geq 9 \epsilon \ln(10)\\ \omega = \Omega(\epsilon)$$ and similarly in the second case $\omega \geq \epsilon \ln (\epsilon)$ so $\omega = \Omega(\epsilon\ln (\epsilon))$