There is logic to be had. Think of "x" as being the process/operation of "multiply by x". Now you want to invert that, so you multiply instead by x^{-1}. Rai

There is logic to be had. Think of "x" as being the process/operation of "multiply by x". Now you want to invert that, so you multiply instead by x^{-1}. Raising to the power of "-1" means that we are inverting the operation in question, so inverting the application of function "f" to argument "x" will be to apply f^{-1} to x instead. Now I'm not saying it's good notation, but some of the fields I've worked in, the notation is not only OK, it's genuinely empowering, as a good notation should be. I have a lot to say about notation in math, but this isn't the forum. There are a lot of crimes committed, agreed, but some of the things people pick out are only bad because they don't know the context where they redeem themselves.