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 theexponential-logarithmic identity that remains largely implicit in the classicalmathematical literature. Rather than treating it solely as a functional relationship betweenthe exponential and logarithmic functions, the article demonstrates that every positive realnumber 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 therepresented numerical value.This interpretation reveals that a single positive real number admits infinitelymany equivalent exponential-logarithmic representations, parametrized by the choice ofbase. By making this representational freedom explicit, the work establishes a systematicframework for the algebraic reorganization of multiplicative expressions in the domainof positive real numbers, shifting their complexity to the additive space of the exponents.The proposed approach does not consist of the classical application of logarithmsas a global operation, but rather of the selective structural reexpression of terms, fullypreserving the numerical identity of the equations involved. As a consequence, it becomespossible to linearize multiplicative relations, homogenize heterogeneous bases, andreorganize additive exponential equations without the need for bilateral transformations.From an algebraic and computational perspective, this property allows for theisolation of irrational constants, the unification of incompatible scales, and the exposureof hidden linear relationships between variables. From a structural perspective, it showsthat the complexity of many problems does not lie in the values themselves, but in theway those values are represented.The applications developed demonstrate that this structural reparametrizationleads to concrete gains in numerical stability, computational efficiency, and analyticalclarity, particularly in contexts involving multiple bases, physical thresholds, or distinctscales. The work thus consolidates the exponential-logarithmic identity as a generalmethodological tool for algebraic reorganization, significantly extending its conceptual,2 analytical, and applied scope, without the introduction of approximations or additionalhypotheses.