Commutativity

1. Take $a=0$ and $b=1$, then $a*_1b=a-b=0-1=-1$ but $b*_1a=b-a=1-0=1\neq a*_1b$ so $*_1$ is not commutative.

2. Take $a=1$ and $b=0$, then $a*_2b=2^{a-b}=\frac{1}{2}$ but $b*_1a=2^{b-a}=2\neq a*_2b$ so $*_2$ is not commutative.

3. This operation is commutative. Indeed, if one consider $a,b\in\mathbb{Q}^*$, then one gets:

$$b*_3a=\frac{b}{a}+\frac{a}{b}=\frac{a}{b}+\frac{b}{a}=a*_3b,$$which is exactly what we wanted !