Are there any known methods which can be used to solve system of equations of the form $$ \begin{align} \ln \frac{P_1(x_1,\ldots,x_n)}{Q_1(x_1,\ldots,x_n)} + R_1(x_1,\ldots,x_n) &= 0 \\ &\ldots \\ \ln \frac{P_n(x_1,\ldots,x_n)}{Q_n(x_1,\ldots,x_n)} + R_n(x_1,\ldots,x_n) &= 0, \end{align} $$ where $P_i(x_1,\ldots,x_n)$, $Q_i(x_1,\ldots,x_n)$ and $R_i(x_1,\ldots,x_n)$ are polynomials, analytically and/or numerically in the positive reals $x_i \in \mathbb{R}_+$?
Perhaps at least in the special case when $P_i(x_1,\ldots,x_n)$ and $Q_i(x_1,\ldots,x_n)$ are linear and solutions are confined to the interior of the standard simplex $x_1 + \ldots + x_n < 1$, $x_i > 0$?
As far as I know, systems of polynomial equations can be solved analytically using Groebner bases, and numerically using homotopy continuation method. Can these methods be adapted for system of equations shown above?
Additional tag: homotopy-continuation
Upon further investigation it appears that, at least, any such system can be considered as a system of complex-analytic equations, which can be solved in a suitably chosen region of $\mathbb{C}^n$ containing standard real simplex numerically via methods from the chapter chapter 4 "Systems of analytic equations" of the book "Computing the Zeros of Analytic Functions" by Peter Kravanja and Marc Van Barel.