Target is to approximate $\frac{1}{\epsilon}\ln (1- \epsilon x) $ ($\epsilon, x \in (0,1) $). Here is one using $\ln (1+y) \approx y $: $$ \frac{1}{\epsilon}\ln (1- \epsilon x) \approx -x $$ I conjecture this is a better approximation although I am not sure how to prove/disprove it: $$ \frac{1}{\epsilon}\ln (1- \epsilon x) \approx - \frac{1}{1-\epsilon} x $$
Any ideas?