The species-area relationship (SAR) is widely applied in ecology. Mathematically it is usually expressed as either a semi-log or power-law relationship, with the former being introduced by Gleason and the latter by Arrhenius. We here resolve the dispute about which form of the SAR to prefer by introducing a novel model that smoothly transforms between the Gleason semi-log (GSL) and Arrhenius power law (APL) forms. The model introduced has the form of lnq(S) = a + z lnA, with lnq being a generalized logarithmic function which is a linear map (y = x) for q = 0 and a logarithmic map (y = ln x) for q = 1 and q can take any intermediate value between 0 and 1. We applied this model to 100 datasets (mostly islands), linking species richness to island area. The APL was the preferred model in 68% of head-to-head comparisons with the GSL. Both models were supported in 40% of cases. In just under half (44%) of the cases an intermediate model best explained the data. The results demonstrate the utility of a simple intermediate SAR model. Visualizing the profile of the range of model fits for all q ∈ [0; 1] allows us to gain extra insight into SARs not yielded by head-to-head comparisons of GSL and APL. The mathematics related to the generalized logarithmic function introduced here promises to have application to other areas of mathematical ecology, for instance in population biology models.