For every pair of complex numbers a and b, provided a is not 1 and neither a nor b is 0, there exists a complex number c such that a^c = b. In other words, there is a number c such that b is the cth power of a. Another way of putting this is that there is a number c such that a is the cth root of b.
Given a and b, we can find c simply by calculating (ln b)/(ln a). Of course (?) the natural log function ln z is multivalued, so there will actually be an infinite number of solutions. If we want just one, we can use the formula c = (Ln b)/(Ln a), where Ln z is the principal value of the complex logarithm.
One related fact is that 0 is not the only value of x that satisfies a^x = 1. For example, 2^(-9.0647202836543876i) = 1. Isn't that remarkable?
ReplyDeleteActually that's a bit sloppy. I should probably have written 2^((-9.0647202836543876...)*i), since the actual number -9.0647... is irrational. To be precise, it's -2pi/ln 2. So in terms of the above formula a^c=b, here we have a=2, b=1, and c=(ln 1)/(ln 2), using -i(2pi) instead of 0 for ln 1. This should work for any other value of ln 1, namely 0 + k(i(2pi)) for any integer k.
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