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In this survey article we guide the reader through the solution of the commutative equation of exponentiation x = y and that of the associative equation of exponentiation x ) = (x). Various characterizations of the integer, rational, real and complex solutions are discussed together with some new results and open directions. The article is supplemented by a detailed and commented bibliography on the history of these equations.

Two centuries of the equations of commutativity and associativity of exponentiation