Here is my thought process:
- We can define subtraction as adding the inverse of a number w.r.t. the addition operation, e.g. $f(x) = x - 2$ is equivalent to taking $x$ and adding to it a number $a$ such that $a + 2 = 0$, and we call $a$ “minus 2”.
- We can define division as multiplying with the inverse of a number w.r.t. the multiplication operation, e.g. $f(x) = \frac{x}{2}$ is equivalent to taking $x$ and multiplying it by a number $a$ such that $a * 2 = 1$, and we call $a$ “1 over 2".
- The same applies to roots as the inverses of raising to a power, although here things already start to get a bit weird, since we can think of $f(x) = \sqrt{x}$ as raising $x$ to a power $a$ such that $a * 2 = 1$. This is peculiar because in order to find the inverse of the raising to a power operation, we make use of the rules of multiplication (which arise from the rule that ${x^a}^b = x^{a * b}$, but still seems peculiar).
- With exponentiation this breaks apart completely. For example, $f(x) = \log_{2}{x}$ cannot be written as “exponentiating” $x$ in any way. It can be explained as taking $2$ and raising it to an exponent containing $x$ such that the end result is $x$, and we call that exponent “the logarithm base 2 of x”. However, this way of explaining things is conceptually different from cases 1-3.
I would assume those observations could be explained through abstract algebra, which would explain why the raising to a power operation relies on the properties of multiplication, and why exponentiation is in a category of its own, but I don’t know enough about this field to make those connections.
