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Complex exponents with a positive real base

If b is a positive real number, exponentiation with base b and complex exponent z is defined by means of the exponential function with complex argument (see the end of § Exponential function, above) as

where ln b {\displaystyle \ln b} denotes the natural logarithm of b.

This satisfies the identity

In general, (b z) t {\textstyle \left(b^{z}\right)^{t}} is not defined, since b is not a real number. If a meaning is given to the exponentiation of a complex number (see § Non-integer powers of complex numbers, below), one has, in general,

unless z is real or t is an integer.

Euler's formula,

allows expressing the polar form of b z {\displaystyle b^{z}} in terms of the real and imaginary parts of z, namely

where the absolute value of the trigonometric factor is one. This results from