How to compute tetration of values where the value $k$ is a negative integer?

Question

I would like to know about how to exactly do calculation with tetration, especially when the value $k$ is a negative value in: $a ↑↑ k$

I am aware of the process of tetration, which is repeated exponentiation. I would like to know the result when $k$ is negative, and how that is possible. Would also be helpful if I could also get an example with this.

Yours Sincerely, Aster17

Answer 1

So,

It seems tetrating values to a negative value seems to be a still studied phenomenon. But, the answer seems to be the following:

$a ↑↑ -1$
$=> log$ _a $ (a^0) $
$=> log$ _a $ (1) $
$=> 0 $

It seems to remain undefined for any negative integer aside from -1, due to the fact that there is no finite number which you can raise a number to that will lead to 0.

So, the answer is undefined for now, until tetration itself gets... you could say, well-defined!

Bad puns aside, I would like to thank Graviton for leading me to the right resources for getting the right answer and I will link them down below for further information.

Cheers,
Aster17

References:
https://math.eretrandre.org/tetrationforum/index.php
https://en.wikipedia.org/wiki/User:MathFacts/Tetration_Summary
How would tetration work for non integer numbers.