Unified Solving of Transcendental Equations with Logarithms via Hierarchical Differential Algebraic Closure

This paper presents a novel theoretical and computational framework for solving transcendental equations involving logarithmic terms through an extension of hierarchical differential algebraic closure (DAC). Building upon our previous work on multivariate polynomial systems, we generalize the differential algebraic approach to equations involving logarithmic terms by constructing a recursively layered closure space incorporating both polynomial and logarithmic differential operators. For a system of m equations f(x,L) = 0 in n variables x = (x1,...,xn) with logarithmic terms L = (log(x1),...,log(xn)), we prove that all solutions can be expressed analytically through an explicit representation theorem involving representation-theoretic components. Our method achieves machine-precision accuracy (residuals < 10−28) with complexity O((de)n) for general systems and O(n2) for sparse symmetric systems, significantly outperforming traditional methods in numerical stability. Extensive validation confirms the robustness, efficiency, and numerical stability of our approach. Furthermore, the proposed framework naturally extends to incorporate generalized symmetries and tensor network representations, opening up novel pathways for hybrid symbolic-numeric computation of transcendental equations.