Let $G=\{x \in \mathbb R \mid x>0 , x\neq 1 \}$. Define $*$ operation on $G$ by $a*b:=a^{\ln(b)}$ for all $a,b \in G$. Prove that $G$ is an abelian group under the operation $*$.
I know that definition of abelian group is like this
A group $G$ is said to be abelian if $a*b=b*a$ for all $a,b \in G$.
But how $a^{\ln b}$ should be equal to $b^{\ln(a)}$? I did not understand this and please help me, if $a*b=a^{\ln(b)}$ then by the same logic is not $b*a=b^{\ln(a)}$?if so how are they equal to each other? Thanks a lot.