  This work falls within the theory of linear forms in logarithms over a
commutative linear group defined over a number field. We give lower bounds for
simultaneous linear forms in logarithms of algebraic numbers, treating both the
archimedean and $p$-adic cases. The proof includes Baker's method, Hirata's
reduction, Chudnovsky's process of variable change. The novelty is that we
integrated into the proof the modern tools of adelic slope theory, using also a
new small values Siegel's lemma.
