using sinh(x) ?

[sheldonison]

Hm, so even there! Actually I found out that you can do the basechange also for bases smaller \( \eta \): The limit
\( \mu_{a,b}(x)=\lim_{n\to\infty} \log_a^{[n]}(\exp_b^{[n]}(x)) \) exists for all \( 1<a<b \) and \( x\in (-\infty,\infty) \) and satisfies \( \mu_{a,b}\circ\exp_b = \exp_a\circ\mu_{a,b} \).
However it is constant for \( x\le a^+ \) (the upper fixed point of of \( a^x \)). BUT it is strictly increasing for \( x>a^+ \)! This means we can use the change of base to change bases of regular tetration at the *upper* fixed point.

I followed most of that, what is the constant value? the upper fixed point of a? Also, what is the circle \( \circ \) notation, \( \mu_{a,b}\circ\exp_b = \exp_a\circ\mu_{a,b} \)? Is it the same as \( \mu_{a,b}(\exp_b) = \exp_a(\mu_{a,b}) \)? No...

But even then:
If \( \sigma_a \) and \( \sigma_b \) are the regular superfunctions to base \( a \) and \( b \) respectively at the *upper* fixed point, then
\( \mu_{a,b}\circ \sigma_b \) is a superfunction to base \( a \) but
\( \mu_{a,b}\circ \sigma_b \neq \sigma_a \).

Yeah, I would expect all of the same problems of iterating complex logarithms, with super exponential windings, and singularities.