Taking the logarithm of $e^{-x}<b$

Question

If I have an inequality as $e^{-x}<b$ where $b,x$ are positive ,
can I take the logarithm on both sides and say,
$-x<=ln(b)$

Answer 1

You even have $$ -x < \operatorname{ln}(b), $$ because $ln \colon (0, \infty) \to \mathbb R$ is strictly increasing on its entire domain.