Show that the set of positive real numbers form a vector spa

Show that the set of positive real numbers form a vector space under the operation x+y=xy and ax=x^a All ten axioms must be verified.

Checking associativity:

x+(y+z)=x+yz=x(yz)=xyz

(x+y)+z=xy+z=(xy)z=xyz

Checking commutativity

x+y=xy=yx=y+x

Identity element

1 forms the identity

x+1=x*1=x

Inverse element

x+x^{-1}=xx^{-1}=1

So each non zero x has an inverse.

Compatibility of scalar multipliciation with field multiplication

a(bx)=ax^b=(x^b)^a=x^(ab)=(ab)x

Identity of multiplication

1*x=x^1=x

Distributivity of scalar multiplication w.r.t vector addition

a(x+y)=a(xy)=(xy)^a=x^ay^a=x^a+y^a=ax+ay

Distributivity of scalar multiplication with respect to field addition

(a+b)x=x^(a+b)=x^ax^b=x^a+x^b=ax+bx