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

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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:

\(\ds \forall x, y \in \R_{>0}: \, \) \(\ds x \oplus y\) \(=\) \(\ds \map f {\map {f^{-1} } x + \map {f^{-1} } y}\)
\(\ds \) \(=\) \(\ds \map f {10^x + 10^y}\)
\(\ds \) \(=\) \(\ds \map {\log_{10} } {10^x + 10^y}\)

$\blacksquare$


Sources