Positive real multiplicative group

The positive real multiplicative Lie group is $\mathbb R_{>0}$ with multiplication, identity 1, and inversion $x\mapsto x^{-1}$ .

Dimensions and structure

This group has real dimension $1$ . Logarithm identifies it with the additive real line.

The logarithm is a smooth group isomorphism, since $\log(xy)=\log x+\log y$ , with inverse the real exponential. See $(\mathbb R^{1},+)$ .

Direct verification

Multiplication is smooth on the positive real numbers, inversion is x ↦ 1/x, and logarithm is a smooth group isomorphism to the additive real line.