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:

\(\ds \forall x, y \in \R_{>0}: \, \)

\(\ds x \otimes y\)

\(=\)

\(\ds \map f {\map {f^{-1} } x \times \map {f^{-1} } y}\)

\(\ds \)

\(=\)

\(\ds \map f {\log_{10} x \times \log_{10} y}\)

\(\ds \)

\(=\)

\(\ds 10^{\log_{10} x \times \log_{10} y}\)

\(\ds \)

\(=\)

\(\ds \paren {10^{\log_{10} x} }^{\log_{10} y}\)

\(\ds \)

\(=\)

\(\ds x^{\log_{10} y}\)

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 6$: Isomorphisms of Algebraic Structures