How to choose a variable make this inequality hold?

Question

Given a integer $n$ and small positive real number $\epsilon$, how to choose $q = q(n,\epsilon)$ such that the following inquality hold asymotopicaly? $$\frac{\log q}{q}\leq \frac{\epsilon}{n}$$

Answer 1

The solution of $$\frac{\log (q)}{q}= \frac{\epsilon}{n}$$ is given by $$q=-\frac n{\epsilon}\,W\left(-\frac{\epsilon }{n}\right)$$ where appears Lambert function.

If we let $k= \frac{\epsilon}{n}$, for small values of $k$ we have $$q=1+k+\frac{3 k^2}{2}+\frac{8 k^3}{3}+O\left(k^4\right)$$