Abstract : The study of linear forms in logarithms is a branch of transcendental number theory, investigating the linear independence of logarithms of algebraic numbers over the field of algebraic numbers $\bar{Q}$. The central tool in the theory is Baker's method, which gives non trivial lower bounds for the absolute value of linear forms in logarithms. This method has been generalized in a much more general setting by Wüstholz and by Philippon and Waldschmidt, by considering logarithms on an arbitrary commutative algebraic group $G$ defined over a number field. One recovers the classical case of the theory when $G$ is linear.

In this talk, I will give an introduction to the theory and present the main steps of Baker's method. I will then describe how to set up the problem in the general case of an arbitrary group, following Philippon and Waldschmidt. I shall emphasize a particular difficulty arising in this general context, usually called the periodic case. Finally, I will explain a new approach to deal with the periodic case and present applications to generalize previous works of Gaudron.