  We determine the mod $p$ reductions of all two-dimensional semi-stable
representations $V_{k,\mathcal{L}}$ of the Galois group of $\mathbb{Q}_p$ of
weights $3 \leq k \leq p+1$ and $\mathcal{L}$-invariants $\mathcal{L}$ for
primes $p \geq 5$. In particular, we describe the constants appearing in the
unramified characters completely. The proof involves computing the reduction of
Breuil's $\mathrm{GL}_2(\mathbb{Q}_p)$-Banach space $\tilde{B}(k,\mathcal{L})$,
by studying certain logarithmic functions using background material developed
by Colmez, and then applying an Iwahori theoretic version of the mod $p$ Local
Langlands Correspondence.
