Transplant (Abstract Algebra)/Examples/Multiplication on Reals under Tenth Power

Example of Transplant
Let $\struct {\R, \times}$ be the set of real numbers under multiplication.

Let $\R_{>0}$ be the set of strictly positive real numbers.

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

The transplant $\otimes$ of $\times$ on $\R$ under $f$ is given by:
 * $\forall x, y \in \R_{>0}: x \otimes y = x^{\log_{10} y}$

Proof
From Group Isomorphism Examples: Real Power Function, $f$ is an isomorphism.

Hence $f$ is a bijection.

The inverse of $f$ is given as:
 * $\forall y \in \R_{>0}: \map {f^{-1} } y = \log_{10} y$

Hence from the Transplanting Theorem: