Commutative lie groups - how is $(R, >, 1)$ a $T^q \times R^p$

Question

I just found out that the connected, commutative lie groups are all products of the form $T^q \times R^p$, where T is the circle and R the real numbers.

Is the set of positive reals under multiplication not also a Lie group? How does this fit into this classification?

I don't know anything about Lie groups, so perhaps there is some very obvious isomorphism that I am missing?

Answer 1

Exponential map, obviously. Never mind.