Linear independence of values of logarithms revisited

Let $m\ge 2$ be an integer, $K$ an algebraic number field and $\alpha\in K\setminus {0,-1}$ with sufficiently small absolute value. In this article, we provide a new lower bound for linear form in $1,\mathrm{l}\mathrm{o}\mathrm{g}(1+\alpha ),\dots ,{\mathrm{l}\mathrm{o}\mathrm{g}}^{m-1}(1+\alpha )$ with algebraic integer coefficients in both complex and $p$-adic cases (see Theorem $2.1$ and Theorem $2.4$). Especially, in the complex case, our result is a refinement of the result of Nesterenko-Waldschmidt on the lower bound of linear form in certain values of power of logarithms. The main integrant is based on Hermite-Mahler’s Pad'{e} approximation of exponential and logarithm functions.