On the regularity of $\{\lfloor {\log }_b(\alpha n+\beta )\rfloor {\}}_{n\ge 0}$

Let $\alpha ,\beta$ be real numbers and $b\ge 2$ be an integer. Allouche and Shallit showed that the sequence $\{\lfloor \alpha n+\beta \rfloor {\}}_{n\ge 0}$ is $b$-regular if and only if $\alpha$ is rational. In this paper, using a base-independent regular language, we prove a similar result that the sequence $\{\lfloor {\log }_b(\alpha n+\beta )\rfloor {\}}_{n\ge 0}$ is $b$-regular if and only if $\alpha$ is rational. In particular, when $\alpha =\sqrt{2},\beta =0$ and $b=2$, we answer the question of Allouche and Shallit that the sequence $\{\lfloor \frac{1}{2}+{\log }_{2}n\rfloor {\}}_{n\ge 0}$ is not $2$-regular, which has been proved by Bell, Moshe and Rowland respectively.