Inequality with little-o notation

Question

I'm having trouble justifying the following: For large $n$, \begin{align*} -\log f(n) & < \log n + o(\log n)\\ \implies f(n) &> n^{-1} \log^3(n) \log(10) \end{align*}

I think basically for large $n$ they claim $e^{-o(\log n)} > \log^3(n) \log(10)$?

Edit: the first inequality should have been strict, corrected

Answer 1

No wonder you have trouble justifying it, because the implication is false. The function $f(n)= n^{-1}$ is a counterexample: it satisfies the first asymptotic relation, but not the second.