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Is anything interesting known about the binary operation $$ x\circ y = \exp_b((\log_b x)(\log_b y)) $$ where $0?

It's clearly commutatitive and associative, and satisfies $\forall x\in \mathbb R^+$, $x\circ b=x$. In one sense is obviously equivalant to multiplication so nothing interesting can be said after everything's been said about multiplication, but then we can wonder if there anything interesting to be said about this way of embedding a structure isomorphic to $(\mathbb R^+,\times)$ into the line. In particular, we have (as I noted yesterday in another thread in this forum) $$ \log_{{}\,p \,\circ\, q\,\circ\, r\,\circ\,\cdots} (w\circ x\circ y\circ\cdots) = (\log_{{}\,p} w)(\log_{{}\,q} x)(\log_{{}\,r} y)\cdots. $$

I know I've seen this function arising in routine stuff, but I can't remember any specifics. So a question is: in which contexts does this operation arise naturally?

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    Your "multiplication" $x \circ y$ doesn't seem to interact well with addition -- certainly $\circ$ is not distributive over $+$. However, there are nice relationships with exponentiation: $(x \circ y)^z = x^z \circ y = x \circ y^z$.2012-10-03
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    Under this operation, it's easy to show that $x^{-1}=b^{1/\log_b x}$. Look at the graph of $(x, x^{-1})$ and compare it with the same graph for ordinary multiplication.2012-10-03
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    There is distribution over ordinary multiplication: $x \circ(y \cdot z) = (x \circ y) \cdot (x \circ z)$.2012-10-03
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    I would call this "conjugation of $\times$ with $\log$" (i.e. something like $x\circ y = \log^{-1}(\log(x)\times\log(y))$).2012-10-03
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    This operation arises in homework assignments in group theory courses. You define $a*b=a^{\log b}$ on some appropriate subset of the reals, and then you ask the students, is it associative? is there an identity element? does each element have an inverse? is it commutative? It's a good exercise, since at first glance you might not expect it to have all these properties.2012-10-04
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    @GerryMyerson : But I think I've also seen this operation arise unintentionally in routine things.2012-10-04

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