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We report measurements of the parity-conserving beam-normal single-spin elastic scattering asymmetries Bn on 12C and 27Al, obtained with an electron beam polarized transverse to its momentum direction. These measurements add an additional kinematic point to a series of previous measurements of Bn on 12C and provide a first measurement on 27Al. The experiment utilized the Qweak apparatus at Jefferson Lab with a beam energy of 1.158 GeV. The average laboratory scattering angle for both targets was 7.7∘, and the average Q2 for both targets was 0.024 37 GeV2 (Q=0.1561 GeV). The asymmetries are Bn=−10.68±0.90(stat)±0.57(syst) ppm for 12C and Bn=−12.16±0.58(stat)±0.62(syst) ppm for 27Al. The results are consistent with theoretical predictions, and are compared to existing data. When scaled by Z/A, the Q dependence of all the far-forward angle (θ<10∘) data from 1H to 27Al can be described by the same slope out to Q≈0.35 GeV. Larger-angle data from other experiments in the same Q range are consistent with a slope about twice as steep.