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.