This article is devoted to a study of the Hardy space H^log(R^d) introduced by Bonami, Grellier, and Ky. We present an alternative approach to their result relating the product of a function in the real Hardy space H¹ and a function in BMO to distributions that belong to H^log based on dyadic paraproducts. We also point out analogues of classical results of Hardy–Littlewood, Zygmund, and Stein for H^log and related Musielak–Orlicz spaces.