asymptotics - Inequality with little-o notation

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