  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
$1,{\rm{log}}(1+\alpha),\ldots,{\rm{log}}^{m-1}(1+\alpha)$ 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.
