classically over the rational numbers, the exponential and logarithm series converge $p$-adically within some open disc of $\mathbb{c}_p$. for function fields, exponential and logarithm series arise naturally from drinfeld modules, which are objects constructed by drinfeld in his thesis to prove the langlands conjecture for $\mathrm{gl}_2$ over function fields. for a ``finite place'' $v$ on such a curve, one can ask if the exp and log possess similar $v$-adic convergence properties. for the most basic case, namely that of the carlitz module over $\mathbb{f}_q[t]$, this question has been long understood. in this talk, we will show the $v$-adic convergence for drinfeld-(hayes) modules on elliptic curves and a certain class of hyperelliptic curves. as an application, we are then able to obtain a formula for the $v$-adic $l$-value $l_v(1,\psi)$ for characters in these cases, analogous to leopoldt's formula in the number field case.