Real Exponents With Negative Bases
Powers of a positive real number are always positive real numbers. The solution of x2 = 4, however, can be either 2 or −2. The principal value of 41/2 is 2, but −2 is also a valid square root. If the definition of exponentiation of real numbers is extended to allow negative results then the result is no longer well behaved.
Neither the logarithm method nor the rational exponent method can be used to define br as a real number for a negative real number b and an arbitrary real number r. Indeed, er is positive for every real number r, so ln(b) is not defined as a real number for b ≤ 0.
The rational exponent method cannot be used for negative values of b because it relies on continuity. The function f(r) = br has a unique continuous extension from the rational numbers to the real numbers for each b > 0. But when b < 0, the function f is not even continuous on the set of rational numbers r for which it is defined.
For example, consider b = −1. The nth root of −1 is −1 for every odd natural number n. So if n is an odd positive integer, (−1)(m/n) = −1 if m is odd, and (−1)(m/n) = 1 if m is even. Thus the set of rational numbers q for which (−1)q = 1 is dense in the rational numbers, as is the set of q for which (−1)q = −1. This means that the function (−1)q is not continuous at any rational number q where it is defined.
On the other hand, arbitrary complex powers of negative numbers b can be defined by choosing a complex logarithm of b.
Read more about this topic: Complex Numbers Exponential, Real Exponents
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