We prove an analogue of Fujino and Mori’s “bounding the denominators” (Theorem 3.1 of Fujino and Mori, J Diff Geom 50:167–188, 2000) in the log canonical bundle formula (see also Theorem 8.1 of Prokhorov and Shokurov, J Algebraic Geom 18(1):151–199, 2009) for Kawamata log terminal pairs of relative dimension one. As an application we prove that for a klt pair (X,Δ) of Kodaira codimension one and dimension at most three such that the coefficients of Δ are in a DCC set \({\mathcal{A}}\), there is a natural number N that depends only on \({\mathcal{A}}\) for which \({\lfloor{N(K_X+\Delta)\rfloor}}\) induces the Iitaka fibration. We also prove a birational boundedness result for klt surfaces of general type.