For a given weight function \(w:{\mathbb R}^d \to (0,\infty)\), the space \(B_{p,q}^s({\mathbb R}^d, w)\) is the collection of all tempered distributions \({\mathbb R}^d)\), with the norm \(\| f| \, B_{p,q}^s({\mathbb R}^d, w)\| = \| fw| \, B_{p,q}^s({\mathbb R}^d\| \). The authors find the order of decay of the entropy numbers of the embeddings \(B_{p,q}^s({\mathbb R}^d, w_1)\to B_{p,q}^s({\mathbb R}^d, w_2)\) under restrictions on the weights \(w_1\) and \(w_2\) that are much weaker then those considered earlier by other authors. The problem is discretized with the help of wavelet decompositions and the corresponding sequence spaces.