The set of positive reals under multiplication is isomorphic to the set of reals under addition, which is the isomorphism underlying the operation of a slide rule.  This is the only isomorphism I can think of important enough that its explicit (approximate) values used to be published in 1000-page books.  The reals under multiplication is also a standard pedagogical example of an interesting one-dimensional abstract real vector space, where there is some content to verifying the axioms.  (The other standard example is the reals with addition given by $x+y-1$ and multiplication by scalar $a$ given by $ax + 1 - a$.)