Diophantine approximation and algebraic independence of logarithms

We prove that any transcendental complex number is well approximated by algebraic numbers of large degree and bounded absolute logarithmic height. Next we extend this result to a statement on simultaneous diophantine approximation for any finite subset of a field of transcendence degree 1 over Q. This tool enables us to introduce a new method for algebraic independence, which we develop in the context of several parameters subgroups of linear algebraic groups. We show for instance that if log alpha(1),...,log alpha(n) are Q-linearly independent logarithms of algebraic numbers in a field of transcendence degree 1 over Q, then for any non zero quadratic form Q is an element of Q[X-1,...,X-n], the number Q(log alpha(1),...,log alpha(n)) does not vanish.