  Let $X$ be an elliptic curve or a ramifying hyperelliptic curve over $\mathbb
F_q$. We will discuss how to factorize the coefficients of the exponential and
logarithm series for a Hayes module over such a curve. This allows us to obtain
$v$-adic convergence results for such exponential and logarithm series, for $v$
a 'finite' prime. As an application, we can show that the $v$-adic Goss
$L$-value $L_v(1, \Psi)$ is log-algebraic for suitable characters $\Psi$.
