The index of multiplicative groups of algebraic numbers

Fix several non-zero elements of an algebraic field with linearly independent logarithms. Consider the set of elements of the field whose logarithms can be expressed in terms of the logarithms of the fixed numbers using rational coefficients. The corresponding vectors of coefficients make up a lattice with the standard integral lattice as a finite-index sublattice. An improved upper bound for this index is given in terms of the extended logarithmic heights of the quantities involved. On the way an estimate for the coefficients in integer linear relations between the logarithms of algebraic numbers is obtained.