A while ago I was thinking about what are known in my area as "the laws of exponentiation" and the "the laws of logarithms", a set of identities for the functions of logarithms and exponentiation. A examples from the set of each include $e^{a+b}=e^{a} \times e^{b}$ and $\ln(a \times b)=\ln(a) + \ln(b)$. I found it interesting to notice that the two functions, $e^x$ and $\ln(x)$, seem to have identities that are in some way the reverse of each other. More specifically, adding the inputs for $e^x$ is equivalent to multiplying the outputs and multiplying the inputs for $\ln(x)$ is equivalent to adding the outputs. The pattern that seems to exist is that the operators, addition and multiplication, swap places between the two identities.
Does a relationship like this exist between all functions' identities and their inverses' identities? If so, what is the relationship in general and how can it be proven?
To my surprise and enjoyment there is a general relationship between the identities, and the proof of such takes only a few steps! First, if $f$ is the function for which the identity is known, its identity can be written as $$f(a \oplus b) = f(a) \otimes f(b)$$ where $\oplus$ and $\otimes$ represent some sort of binary operators. The other thing needed to construct proof is the the fact that $f$'s inverse, $f^{-1}$, by definition relates to it so that $f(f^{-1}(x))=f^{-1}(f(x))=x$. The proof follows and starts with the given identity for $f$, $$f(a \oplus b) = f(a) \otimes f(b)$$ applying the inverse to both sides results in $$f^{-1}(f(a \oplus b)) = f^{-1}(f(a) \otimes f(b))$$ which by the defining relationship of $f^{-1}$ and $f$ leads to $$a \oplus b = f^{-1}(f(a) \otimes f(b)).$$ defining two new variables $c$ and $d$ as $c=f(a)$ and $d=f(b)$ allows the previous to be rewritten as $$f^{-1}(c) \oplus f^{-1}(d)=f^{-1}(c \otimes d)$$ since, using the defining relationship of $f^{-1}$ again, $f^{-1}(a)=c$ and $f^{-1}(b)=d$.
With that final line of $f^{-1}(c) \oplus f^{-1}(d)=f^{-1}(c \otimes d)$ the relationship hypothesized was proven.
The proof can also be generalized to situations where the identity involves any number variables. The only change is that the identity is written as $$f\left(\bigoplus x_n\right) = \bigotimes f(x_n)$$ where $\bigoplus$ and $\bigotimes$ are operators that take $n$ inputs (that is, have arity $n$) and $x_n$ are a set of $n$ variables. The proof's steps themselves are $$f\left(\bigoplus x_n\right) = \bigotimes f(x_n)$$ $$f^{-1}\left(f\left(\bigoplus x_n\right)\right) = f^{-1}\left(\bigotimes f(x_n)\right)$$ $$\bigoplus x_n = f^{-1}\left(\bigotimes f(x_n)\right)$$ ($y_n=f(x_n), f^{-1}(y_n)=x_n$) $$\bigoplus f^{-1}(y_n) = f^{-1}\left(\bigotimes y_n\right)$$