How can you solve for an unknown operator and investigate its properties?

[MWresearch]

I'm not solving a differential equation with an unknown function but it seems it should be similar. What I mean is I want to solve for an unknown operator that has some arbitrary transformation of terms and variables. If for instance I had x^5 and I wanted to find some operator that made x^5 turn into x^4, I should get ^4/5. But let's say I didn't know that property of exponents as a given, how would I mathematically investigate the apparent transformation of variables to find that operation that has the properties of turning x^5 into x^4?

Edited September 16, 2015 by MWresearch