Automorphism group of $(mathbb R_{>0}, cdot)$

I'm trying to find the automorphism group of $(\mathbb R_{>0}, \cdot)$.

What I have so far:

$x \mapsto x^r$ for any $r \in \mathbb R - \{0\}$ is an automorphism.

But I can't think of any others. I have found $|\mathbb R|$ many but because I think $\text{Aut} (\mathbb R_{>0})$ should be much larger than $\mathbb R$ I believe that I didn't find them all.

Which automorphisms am I missing?