Various dark matter models predict annual and diurnal modulations of dark matter interaction rates in Earth-based experiments as a result of the Earths motion in the halo. Observation of such features can provide generic evidence for detection of dark matter interactions. This paper reports a search for both annual and diurnal rate modulations in the LUX dark matter experiment using over 20 calendar months of data acquired between 2013 and 2016. This search focuses on electron recoil events at low energies, where leptophilic dark matter interactions are expected to occur and where the DAMA experiment has observed a strong rate modulation for over two decades. By using the innermost volume of the LUX detector and developing robust cuts and corrections, we obtained a stable event rate of 2.3$pm$0.2~cpd/keV$_{text{ee}}$/tonne, which is among the lowest in all dark matter experiments. No statistically significant annual modulation was observed in energy windows up to 26~keV$_{text{ee}}$. Between 2 and 6~keV$_{text{ee}}$, this analysis demonstrates the most sensitive annual modulation search up to date, with 9.2$sigma$ tension with the DAMA/LIBRA result. We also report no observation of diurnal modulations above 0.2~cpd/keV$_{text{ee}}$/tonne amplitude between 2 and 6~keV$_{text{ee}}$.
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