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?