Real Exponents
The identities and properties shown above for integer exponents are true for positive real numbers with non-integer exponents as well. However the identity
cannot be extended consistently to where b is a negative real number, see Real exponents with negative bases. The failure of this identity is the basis for the problems with complex number powers detailed under failure of power and logarithm identities.
The extension of exponentiation to real powers of positive real numbers can be done either by extending the rational powers to reals by continuity, or more usually by using the exponential function and its inverse, the natural logarithm.
Read more about this topic: Exponentiation
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