text: Failure of power and logarithm identities
  Some identities for powers and logarithms for positive real numbers will fail for complex numbers, no matter how complex powers and complex logarithms are defined as single-valued functions. For example:
  Irrationality and transcendence
  If  is a positive real algebraic number, and  is a rational number, then  is an algebraic number. This results from the theory of algebraic extensions. This remains true if  is any algebraic number, in which case, all values of  (as a multivalued function) are algebraic. If  is irrational (that is, not rational), and both  and  are algebraic, Gelfond–Schneider theorem asserts that all values of  are transcendental (that is, not algebraic), except if  equals  or .
  In other words, if  is irrational and  then at least one of ,  and  is transcendental.
  Integer powers in algebra
  The definition of exponentiation with positive integer exponents as repeated multiplication may apply to any associative operation denoted as a multiplication. The definition of  requires further the existence of a multiplicative identity.
  An algebraic structure consisting of a set together with an associative operation denoted multiplicatively, and a multiplicative identity denoted by  is a monoid. In such a monoid, exponentiation of an element  is defined inductively by