A comparative study on some representative parametrizations for the total and elastic cross-sections as a function of energy is presented. The dataset comprises pp and scattering in the c.m. energy interval 5 GeV-8 TeV. The parametrization for the total cross-section at low and intermediate energies follows the usual reggeonic structure (non-degenerate trajectories). For the leading high-energy pomeron contribution, we consider three distinct analytic parametrizations: either a power (P) law, or a log-squared (L2) law or a log-raised-to-γ (Lγ) law, where the exponent γ is treated as a real free fit parameter. The parametrizations are also extended to fit the elastic (integrated) cross-section data in the same energy interval. Our main conclusions are the following: (1) the data reductions with the logarithmic laws show strong dependence on the unknown energy scale involved, which is treated here either as a free parameter or fixed at the energy threshold; (2) the fit results with the P law, the L2 law (free scale) and the Lγ law (fixed scale and exponent γ above 2) are all consistent within their uncertainties and with the experimental data up to 7 TeV, but they partially underestimate the high-precision TOTEM measurement at 8 TeV; (3) once compared with these results, the L2 law with fixed scale is less consistent with the data and, in the case of a free scale, this pomeron contribution decreases as the energy increases below the scale factor (which lies above the energy cutoff); (4) in all cases investigated, the predictions for the asymptotic ratio between the elastic and total cross-sections, within the uncertainties, do not exceed the value 0.430 (therefore, below the black-disc limit) and the results favor rational limits between 1/3 and 2/5. 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