Lower bounds for linear forms
\Lambda=b_1\log\alpha_1+...+b_n\log\alpha_n, (b_j, \alpha_j\in\overline Q),
have numerous applications to different problems of Number Theory. Usually these bounds for |\Lambda| are expressed in terms of the heights h(\alpha_j), in b_j and of the degree of the field K generated by the numbers \alpha_j. After a long line of results following A.O.Gelfond (1935-1949, n=2) and A.Baker (1966, n\geq 3) the dependence of the bounds in these parameters is practically established. In a recent article (1998) E.Matveev has improved the dependence in terms of n. In the special case where
\deg_K K(\alpha_1^{1/2},...,\alpha_n^{1/2})=2^n {1}
he reduced a factor n^n in the bound to C^n, where C is an absolute constant. Now he proved such result in the general case without the extra condition {1}. This proof is a subject of the talk.