Transplant (Abstract Algebra)/Examples/Addition on Positive Reals under Log Base 10

Example of Transplant
Let $\struct {\R_{>0}, +}$ be the set of strictly positive real numbers under addition.

Let $f: \R_{>0} \to \R$ be the permutation defined as:
 * $\forall x \in \R_{>0}: \map f x = \log_{10} x$

The transplant $\oplus$ of $+$ under $f$ is given by:
 * $x \oplus y = \map {\log_{10} } {10^x + 10^y}$

Proof
From Logarithm on Positive Real Numbers is Group Isomorphism, $f$ is a bijection.

The inverse of $f$ is given as:
 * $\forall x \in \R: \map {f^{-1} } x = 10^x$

Hence from the Transplanting Theorem: