Parametric Property of the Exponential-Logarithmic Identity: Reparametrization by an Arbitrary Base and Structural Linearization in the Domain of the Positive Real Numbers

This work makes explicit and systematizes a structural property of the exponential-logarithmic identity that remains largely implicit in the classical mathematical literature. Rather than treating it solely as a functional relationship between the exponential and logarithmic functions, the article demonstrates that every positive real number can be represented exactly in the form R = B^(logB R), for any admissible base, thereby rendering the base a free parameter of reparametrization without altering the represented numerical value. This interpretation reveals that a single positive real number admits infinitely many equivalent exponential-logarithmic representations, parametrized by the choice of base. By making this representational freedom explicit, the work establishes a systematic framework for the algebraic reorganization of multiplicative expressions in the domain of positive real numbers, shifting their complexity to the additive space of the exponents. The proposed approach does not consist of the classical application of logarithms as a global operation, but rather of the selective structural reexpression of terms, fully preserving the numerical identity of the equations involved. As a consequence, it becomes possible to linearize multiplicative relations, homogenize heterogeneous bases, and reorganize additive exponential equations without the need for bilateral transformations. From an algebraic and computational perspective, this property allows for the isolation of irrational constants, the unification of incompatible scales, and the exposure of hidden linear relationships between variables. From a structural perspective, it shows that the complexity of many problems does not lie in the values themselves, but in the way those values are represented. The applications developed demonstrate that this structural reparametrization leads to concrete gains in numerical stability, computational efficiency, and analytical clarity, particularly in contexts involving multiple bases, physical thresholds, or distinct scales. The work thus consolidates the exponential-logarithmic identity as a general methodological tool for algebraic reorganization, significantly extending its conceptual, analytical, and applied scope, without the introduction of approximations or additional hypotheses.