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 e1/e, 0 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.
shifa liu (Wed,) studied this question.