text: The limit that defines the exponential function converges for every complex value of , and therefore it can be used to extend the definition of , and thus  from the real numbers to any complex argument . This extended exponential function still satisfies the exponential identity, and is commonly used for defining exponentiation for complex base and exponent.
  Powers via logarithms
  The definition of  as the exponential function allows defining  for every positive real numbers , in terms of exponential and logarithm function. Specifically, the fact that the natural logarithm  is the inverse of the exponential function  means that one has
  for every . For preserving the identity  one must have
  So,  can be used as an alternative definition of  for any positive real . This agrees with the definition given above using rational exponents and continuity, with the advantage to extend straightforwardly to any complex exponent.
  Complex exponents with a positive real base
  If  is a positive real number, exponentiation with base  and complex exponent  is defined by means of the exponential function with complex argument (see the end of , above) as
  where  denotes the natural logarithm of .
  This satisfies the identity
  In general,
  is not defined, since  is not a real number. If a meaning is given to the exponentiation of a complex number (see , below), one has, in general,
  unless  is real or  is an integer.
  Euler's formula,
  allows expressing the polar form of  in terms of the real and imaginary parts of , namely