The boundedness and compactness of a product-type operator, recently introduced by S. Stević, A. Sharma and R. Krishan, Tnψ1 ,ψ2 ,ϕ f (z) = ψ1(z) f (n)(ϕ(z)) + ψ2(z) f (n+1)(ϕ(z)), f ∈ H(D), from the logarithmic Bloch spaces to Zygmund-type spaces are characterized, where ψ1, ψ2 ∈ H(D), ϕ is an analytic self-map ofD and n a positive integer