Lojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations

We give an expression for the {\L}ojasiewicz exponent of a wide class of n-tuples of ideals $(I_1,\mathrm{.}\mathrm{.}\mathrm{.},I_n)$ in ${\text{O}}_n$ using the information given by a fixed Newton filtration. In order to obtain this expression we consider a reformulation of {\L}ojasiewicz exponents in terms of Rees mixed multiplicities. As a consequence, we obtain a wide class of semi-weighted homogeneous functions $({\mathbb{C}}^n,0)\to (\mathbb{C},0)$ for which the {\L}ojasiewicz of its gradient map $\mathrm{\nabla }f$ attains the maximum possible value.