Classical Diophantine approximation studies how fast can we approach real numbers by rationals. A variant of this problem is to restrict the denominators of the approximating rationals to a subset of the natural numbers. In this talk, I will consider approximation by $b$-ary rationals where $b$ is a fixed natural number at least $2$. First, I will present a result relating approximation properties with respect to one base with asymptotic properties with respect to other bases. Afterwards, I will present some observations in the spirit of Lochs' Theorem in order to motivate a new exponent of approximation similar to the ones studied by Amou-Bugeaud and Bugeaud-Liao. This talk is based on a collaboration with Mumtaz Hussain and Simon Kristensen and with Mumtaz Hussain, Nikita Shulga, and Zhengliang Zhang.