Commutative exponentiation

In mathematics, commutative exponentiation is an extension of exponentiation that has the commutative property. Addition and multiplication have this property, but exponentiation does not. Commutative exponentiation extends the commutative property to exponentiation by noting the hierarchy (addition, multiplication, exponentiation) and using the fact that the logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers. It answers the question, "If multiplication is the exponent of the sum of logarithms, then what is the exponent of the product of logarithms?"

A ᐤ B = A ^{ln B} = B ^{ln A} = e^{ln(A) · ln(B)}

where "ᐤ" is the symbol used in this article for commutative exponentiation.

Algebraic properties

Commutative exponentiation has the following algebraic properties:

1. commutation: A ᐤ B = e^{ln(A) · ln(B)} = B ᐤ A
2. association: A ᐤ (B ᐤ C) = (A ᐤ B) ᐤ C
3. distribution over multiplication: C ᐤ (A · B) = e^{ln(C) · (ln(A) + ln(B))} = C ᐤ A · C ᐤ B
4. identity: A ᐤ e = e ᐤ A = A
5. inverse: A ᐤ A⁻¹ = e, where the inverse may be computed: A⁻¹ = e^{1 / ln(A)}

Hyperoperation

Commutative exponentiation is a commutative hyperoperation. "Commutative hyperoperations" were considered by Albert Bennett as early as 1914, which is possibly the earliest remark about any hyperoperation sequence." See Hyperoperation.

See also

• Algebra
• Hyperoperation