text: Real exponents For positive real numbers, exponentiation to real powers can be defined in two equivalent ways, either by extending the rational powers to reals by continuity (, below), or in terms of the logarithm of the base and the exponential function (, below). The result is always a positive real number, and the identities and properties shown above for integer exponents remain true with these definitions for real exponents. The second definition is more commonly used, since it generalizes straightforwardly to complex exponents.
  On the other hand, exponentiation to a real power of a negative real number is much more difficult to define consistently, as it may be non-real and have several values. One may choose one of these values, called the principal value, but there is no choice of the principal value for which the identity
  is true; see . Therefore, exponentiation with a basis that is not a positive real number is generally viewed as a multivalued function.
  Limits of rational exponents
  Since any irrational number can be expressed as the limit of a sequence of rational numbers, exponentiation of a positive real number  with an arbitrary real exponent  can be defined by continuity with the rule
  where the limit is taken over rational values of  only. This limit exists for every positive  and every real .