Editing Exponentiation (section)

==Complex exponents with a positive real base==
If {{mvar|b}} is a positive real number, exponentiation with base {{mvar|b}} and [[complex number|complex]] exponent {{mvar|z}} is defined by means of the exponential function with complex argument (see the end of ''{{slink||Exponential function}}'', above) as
: <math>b^z = e^{(z\ln b)},</math>
where <math>\ln b</math> denotes the [[natural logarithm]] of {{mvar|b}}.

This satisfies the identity
: <math>b^{z+t} = b^z b^t,</math>
In general,
<math DISPLAY=inline>\left(b^z\right)^t</math> is not defined, since {{math|''b''<sup>''z''</sup>}} is not a real number. If a meaning is given to the exponentiation of a complex number (see ''{{slink||Non-integer powers of complex numbers}}'', below), one has, in general,
: <math>\left(b^z\right)^t \ne b^{zt},</math>
unless {{mvar|z}} is real or {{mvar|t}} is an integer.

[[Euler's formula]],
: <math>e^{iy} = \cos y + i \sin y,</math>
allows expressing the [[polar form]] of <math>b^z</math> in terms of the [[real and imaginary parts]] of {{mvar|z}}, namely
: <math>b^{x+iy}= b^x(\cos(y\ln b)+i\sin(y\ln b)),</math>
where the [[absolute value]] of the [[trigonometry|trigonometric]] factor is one. This results from
: <math>b^{x+iy}=b^x b^{iy}=b^x e^{iy\ln b} =b^x(\cos(y\ln b)+i\sin(y\ln b)).</math>