Tetration is harder to define than the other three main operations because exponentiation is unique from addition and multiplication. The main thing that affects Tetration is there are two inverse functions of exponentiation, unlike addition and multiplication. This means that you can have two different answers for certain questions when only one can be correct. The part I'm having trouble defining wasn't when the base is the unknown variable, but when the 'tetrate' (like the exponent variable in exponentiation) is the unknown variable. If the 'tetrate' is a fraction (or mixed fraction) but the numerator was not 1, I had a lot of trouble since it gave me more than one answer. Then it hit me: I could simplify the fraction so that the numerator was one! the only problem with that was that the denominator then had a decimal in it, which I am currently trying to find a way to represent.
Another thing: I saw the linear approximation of tetration, but it's not going to be 100% accurate. For instance, if you use a super-root (the first inverse of tetration, which is basically exponentiation's root inverse for tetration) with a base of infinity on a number that's not 1, you will not end up with 1. For example, take 'e'. If you take a base infinity super root and use it on 'e', you will end up with e^(1/e), which is not 1. Therefore, not only will the graph not be linear, it won't even be whole! An example of a graph that's not whole (in the way I'm describing it) is floor(x), since you can clearly see that it is in steps, but those steps aren't connected by lines.
I will post more on this later, but if anyone has anything to say about tetration feel free to post a reply.