Why not use the entire? Because the behavior is not consistent. It seems stranger to imagine a function that has singularities for only real bases , then suddenly it becomes free of them and entire for all . It's two wildly different analytical behaviors and that doesn't make much sense. Ideally it would be nice to be able to interpret the tetration at to be what you'd get if you did an analytical continuation in the base from through the complex plane. Finally, at the integer towers of this base, we are still dealing with real numbers: when we take the log of for to get as 0, we are still using a real logarithm of a real number to a real base. So why not continue using this principal real logarithm for the rest of the tet function at this base?
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Solving tetration for base 0 < b < e^e

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