Exponential factorizations of holomorphic maps

We show that any element of the special linear group SL2(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 SL2(C) is not surjective. Our result extends to the linear group GL2(R), where it is sharp as well.