Subtraction and division can be expressed with multiplication and exponentiation, as follows:
a - b = a + (b * -1)
a / b = a * (b ^ -1)
My question is: does this pattern generalize? Can we express logarithms with tetration?
log a b = a ^ (b ^^ -1)
Where log, ^ and ^^ are suitable definitions of logarithm, exponentiation and tetration. To be clear, the formula above, as stated, is obviously wrong given the usual definition of these functions, but what I fundamentally want to know is if there is a sensible formulation of the hyper-operation sequence such that the inverse of any hyper-operation can be expressed as a function of a 1-tier higher hyper-operation.
The definitions of $\log$ and $\exp$ are pretty set in stone, so the main way to salvage your desired identity would be by carefully defining tetration, since the normal definition only works for nonnegative integers anyway.
In particular, if we want $\log_b(a) = a^{b \uparrow \uparrow (-1)}$ then we need $$b \uparrow \uparrow (-1) = \log_a\big( \log_b(a) \big).$$
That's not possible no matter how we define tetration against negative numbers: $b \uparrow \uparrow (-1)$ ought to at least be a function purely of $b$, but this expression depends on $a$ too. Therefore it's impossible to redefine tetration to make this pattern of identities continue.