X^1 = 1. ...
X^2 = 2. ...
X^3 = 3. ...
etc
Or will it always be the case that the exponent and the integer part of the result diverge?
@binkle No: the sequence on the left grows exponentially (growing by a factor of X every time), the sequence on the right grows at most linearly.
@binkle @koantig
You can bound the sequence on the right hand side above by the sequence {n+1} (and below by {n}), so now you're just comparing a geometric sequence {x^n} to a basic arithmetic ("linear") sequence. We know it doesn't work for x<=1. If you want x^n <= n+1 for all n with x>1 (a necessary condition), then you'd need logx <= (1/n)log(n+1). The left hand side is fixed positive number (since x>1) and the right hand side has a limit of 0 (use L'hospitals rule).
For complex numbers exponentiation can be visualized with Euler's formula. The length of the vector follows the same pattern as in real numbers, and you multiply the angle by n to rotate it around the origin.
