Exponential Multiplication: A Mathematical Synthesis

This paper introduces and rigorously analyzes a binary operation called exponential multiplication, defined as a ⊗ b = a · 2^b for a, b ∈ ℝ. This operation provides a formal framework for understanding growth through doubling, yielding the notable result that 1 ⊗ 1 = 2. The algebraic properties of this operation are established, including its identity elements, absorption properties, non-commutativity, non-associativity, and relationship to logarithmic transformations. The algebraic structure is classified as a magma and the inverse operation is derived with full domain analysis.