In this paper we investigate the efficiency of the function field sieve to compute discrete logarithms in the finite fields $\mathbb{F}_{3^n}$. Motivated by attacks on identity based encryption systems using supersingular elliptic curves, we pay special attention to the case where n is composite. This allows us to represent the function field over different base fields. Practical experiments appear to show that a function field over $\mathbb{F}_3$ gives the best results.