We study a new model of Energy-Momentum Squared Gravity (EMSG), called Energy-Momentum Log Gravity (EMLG), constructed by the addition of the term f(T mu nu T mu nu)=alpha ln(lambda T mu nu T mu nu)\d, envisaged as a correction, to the Einstein-Hilbert action with cosmological constant Lambda. The choice of this modification is made as a specific way of including new terms in the right-hand side of the Einstein field equations, resulting in constant effective inertial mass density and, importantly, leading to an explicit exact solution of the matter energy density in terms of redshift. We look for viable cosmologies, in particular, an extension of the standard Lambda CDM model. EMLG provides an effective dynamical dark energy passing below zero at large redshifts, accommodating a mechanism for screening Lambda in this region, in line with suggestions for alleviating some of the tensions that arise between observational data sets within the standard Lambda CDM model. We present a detailed theoretical investigation of the model and then constrain the free parameter alpha ', a normalisation of alpha, using the latest observational data. The data does not rule out the Lambda CDM limit of our model (alpha '=0), but prefers slightly negative values of the EMLG model parameter (alpha '=-0.032 +/- 0.043), which leads to the screening of Lambda. We also discuss how EMLG relaxes the persistent tension that appears in the measurements of H0 within the standard Lambda CDM model.