On the regularity of {⌊logb⁡(αn+β)⌋}n≥0

Let α,β be real numbers and b≥2 be an integer. Allouche and Shallit showed that the sequence {⌊αn+β⌋}n≥0 is b-regular if and only if α is rational. In this paper, using a base-independent regular language, we prove a similar result that the sequence {⌊logb⁡(αn+β)⌋}n≥0 is b-regular if and only if α is rational, where α>0,β≥0. In particular, when α=2,β=0 and b=2, we answer a question of Allouche and Shallit that the sequence {⌊12+log2⁡n⌋}n≥1 is not 2-regular, which has previously been proved by Bell, Moshe and Rowland respectively.