The aim of this book is to serve as an introductory text to the theory of linear forms in the logarithms of algebraic numbers, with a special emphasis on a large variety of its applications. The author intends to help students and researchers to learn what is hidden inside the blackbox “Baker's theory of linear forms in logarithms” (in complex or in \(p\)-adic logarithms) and how this theory applies to many Diophantine problems, including the effective resolution of Diophantine equations, the \(abc\)-conjecture, and upper bounds for the irrationality measure of some real numbers.

Written for a broad audience, this accessible and self-contained book can be used for graduate courses (some 30 exercises are supplied). Specialists will appreciate the inclusion of over 30 open problems and the rich bibliography of over 450 references.