  We show that any element of the special linear group $SL_2(R)$ is a product
of two exponentials if the ring $R$ is either the ring of holomorphic functions
on an open Riemann surface or the disc algebra. This is sharp: one exponential
factor is not enough since the exponential map corresponding to
$SL_2(\mathbb{C})$ is not surjective. Our result extends to the linear group
$GL_2(R)$.
