Operational Mathematics of Exponentiation and Logarithm: A Complete Extension of the Theory of Iteration Counts to the Exponential-Logarithm Pair as Mutual Inverses

This paper systematically extends the theoretical framework of Operational Mathematics—as established in the companion work Operational Mathematics: A Theory of Extending the Number Field of Operation Counts—to exponentiation (the operation of order n =2) and logarithm (its unique inverse operation). Following the identical axiomatic structure, proof methodology, and analytic construction techniques of the original work, we rigorously establish the complete theory of integer-order, rational-order, real-order, and complex-order iterations for the exponential function expa(x) = ax and the logarithmic function loga(x). We introduce six specialized axioms (E1–E6) that capture the essential features of exponentiation and logarithm—the exponential addition theorem, the logarithmic multiplication theorem, mutual invertibility, change-of-base formulas, and their characterizations via differential and integral equations—and prove their compatibility and independence within the general six-axiom framework of Operational Mathematics. The Schröder equation and Abel equation are employed to construct fractional iterative roots for hyperbolic, repelling, parabolic, and indifferent (Siegel disk) fixed points, with a complete unification of both constructions via a rigorous limiting process. The Kneser construction is fully generalized to real-order exponentiation for all base regimes: 1 < a < e1/e, a = e1/e, a > e1/e, 0 < a < e−e, and e−e ≤ a < 1, including the Feigenbaum period-doubling cascade for λ < −1. Analytic continuation to complex iteration counts is established, with a comprehen sive analysis of singularity structure—logarithmic branch points at negative integers (for exponentiation) and positive integers (for logarithm)—and the construction of infinite-sheeted Riemann surfaces whose covering groups are isomorphic to Z. The Carathéodory kernel convergence and boundary behavior at the natural boundary (−∞,−2] are rigorously characterized. We prove that fractional calculus and fractional calculus of variations are special cases of the exponential-logarithm iteration semigroup, satisfying the identical semigroup property OαOβ = Oα+β, with complete spectral theory via Mittag-Leffler eigenfunctions. A categorical equivalence Expa ∼= Logopa is rigorously established with explicit natural isomorphisms, demonstrating that the exponential-logarithm duality is a concrete realization of the general duality between the mathematics of numbers and the mathematics of operations. The structural isomorphism between the n = 2 (exponentiation) and n = 3 (tetration) theories is fully characterized via the level-shift functor. Within the hyperoperation spectrum H(r,a,b), exponentiation (r = 3 in the successor-indexed hierarchy) serves as the pivotal bridge connecting discrete lower hyperoperations to continuous higher ones, with logarithm as its inverse pivot. The complete classification of fixed points for all a > 0 via the Lambert W function is provided, including complex fixed points and their multipliers. High-precision numerical algorithms with explicit Faà di Bruno recurrence for Taylor coefficients are developed, and exponential convergence of order O(ρN) with ρ ≈ 0.5 is proved, along with complete numerical stability analysis and optimal complexity lower bounds. Applications to the Wilson renormalization group (with rigorous critical exponent derivation and universality), anomalous diffusion, quantum mechanics, information geometry (Legendre transform correspondence), integral transforms (fractional Laplace, Fourier, Mellin, and Hankel), Lie groups (Baker-Campbell-Hausdorff formula for vector field commutators), and transcendence theory (conditional and unconditional results with Schanuel’s conjecture) are explored. All previously open problems within this domain are transformed into rigorously proved theorems.