The group you described does not inherit any interesting structure from exponentiation -- indeed, one can easily see that it is isomorphic to the multiplicative

The group you described does not inherit any interesting structure from exponentiation -- indeed, one can easily see that it is isomorphic to the multiplicative group of reals. You could similarly construct a group isomorphic to an additive group of reals. This is an example of the fact that real exponentiation connects different aspects of real numbers, as well as the fact that just abstract algebra language is not enough to express properties of real numbers. You need to somehow relate the algebraic structure of reals to a topologic one, which stems from order imposed on reals and its continuity.