An inequality for a linear form in the logarithms of algebraic numbers

Let ln ₁, ..., ln _m–1 be the logarithms of fixed algebraic numbers which are linearly independent over the field of rational numbers, b₁, ..., b_m–1 rational integers, > 0. A bound from below is deduced for the height of the algebraic number _m under the condition that ¦b₁ ln ₁+...+_bm–1ln _m– ¦ < exp {–H},H=max ¦ b_k ¦ >0.Translated from Matematicheskie Zametki, Vol. 5, No. 6, pp. 681–689, June, 1969.