In this talk, I will present a computation of the image of the Hodge-Tate logarithm map (defined by Heuer) in the case of smooth Stein varieties. When the variety is the affine space, Heuer has proved that this image is equal to the group of closed differential forms. In general, we will see that the image always contains such forms but the quotient can be non-trivial: it contains a Zp-module that maps, via the Bloch-Kato exponential map, to integral classes in the proétale cohomology. This is based on a joint work with V. Ertl and W. Niziol.