We establish inequalities of Ulyanov-type for moduli of smoothness relating the source Lorentz-Zygmund space Lp,r(logL)α−γ,γ>0, and the target space Lp∗,s(logL)α over Rn if 1<p<p∗<∞ and over Tn if 1<p≤p∗<∞. The stronger logarithmic integrability (corresponding to Lp∗,s(logL)α ) is balanced by an additional logarithmic smoothness