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Mapping the expansion history of the universe is a compelling task of physical cosmology, especially in the context of the observational evidence for the recent acceleration of the universe, which demonstrates that canonical theories of cosmology and particle physics are incomplete and that there is new physics still to be discovered. Cosmography is a phenomenological approach to cosmology, where (with some caveats) physical quantities are expanded as a Taylor series in the cosmological redshift $z$, or analogous parameters such as the rescaled redshift $y=z/(1+z)$ or the logarithmic redshift $x=\ln{(1+z)}$. Moreover, the redshift drift of objects following cosmological expansion provides a model-independent observable, detectable by facilities currently under construction, {\it viz.} the Extremely Large Telescope and the Square Kilometre Array Observatory (at least in its full configuration). Here we use simulated redshift drift measurements from the two facilities to carry out an assessment of the cosmological impact and model discriminating power of redshift drift cosmography. We find that the combination of measurements from the two facilities can provide a stringent test of the $Λ$CDM paradigm, and that overall the logarithmic based expansions of the spectroscopic velocity drift are the most reliable ones, performing better than analogous expansions in the redshift or the rescaled redshift: the former nominally gives the smaller error bars for the cosmographic coefficients but is vulnerable to biases in the higher order terms (in other words, it is only reliable at low redshifts), while the latter always performs poorly.