According to the GEVAG (Generalized Entropy Varying-G) framework, any modification to the Bekenstein-Hawking area law would also lead to a varying-$G$ gravity theory in which the effective gravitational constant $G_\text{eff}$ becomes area-dependent. Among a myriad of generalized entropy functions explored in the literature, of special interest is the logarithmic correction of quantum gravity. In this work, we apply GEVAG to investigate the effect of logarithmic correction on very early-time cosmology, including the conditions for inflation. We found that if the coefficient of the logarithmic correction term is negative, $G_\text{eff}$ becomes twice that of the current value; whereas, a positive coefficient leads to a very small value of $G_\text{eff}$, which may ameliorate the "arrow of time" problem. In fact, slow-roll inflation becomes more natural in the latter case. We make some comparisons with the constant-$G$ approach and reveal the advantages of the GEVAG approach. For example, it can evade the sudden singularity that could otherwise arise when the coefficient of the logarithmic correction term is negative. We also check the validity of the generalized second law and comment on the range of the various parameters.