$B'$

Let $n\ge 2$ be an integer and $\alpha_1,\ldots,\alpha_n$ be non-zero algebraic numbers. Let $b_1,\ldots,b_n$ be integers with $b_n\not=0$, and set $B=\max\{3,|b_1|,\ldots,|b_n|\}$. For $j=1,\ldots,n$, set $h^*(\alpha_j)=\max\{h(\alpha_j),1\}$, where $h$ denotes the (logarithmic) Weil height. Assume that the quantity $\Lambda=b_1\log\alpha_1+\cdots+b_n\log\alpha_n$ is nonzero. A typical lower bound of $\log|\Lambda|$ given by Baker's theory of linear forms in logarithms takes the shape $$ \log|\Lambda|\ge-c(n, D)\,h^*(\alpha_1)\cdots h^*(\alpha_n)\log B,\\[-6pt] $$ where $c(n,D)$ is positive, effectively computable and depends only on $n$, and on the degree $D$ of the field generated by $\alpha_1,\ldots,\alpha_n$. However, in certain special cases and in particular when $|b_n|=1$, this bound can be improved to $$ \log|\Lambda|-c(n,D)\,h^*(\alpha_1)\cdots h^*(\alpha_n)\log\frac{B}{h^*(\alpha_n)}.\\[-6pt] $$ The term $B/h^*(\alpha_n)$ in place of $B$ originates in works of Feldman and Baker and is a key tool for improving, in an effective way, the upper bound for the irrationality exponent of a real algebraic number of degree at least $3$ given by Liouville's theorem. We survey various applications of this refinement to exponents of approximation evaluated at algebraic numbers, to the $S$-part of some integer sequences, and to Diophantine equations. We conclude with some new results on arithmetical properties of convergents to real numbers.