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