Chapter 4.2, Problem 78E

Let ${ℝ}^{+}$ be the set of positive real numbers. On ${ℝ}^{+}$ we define the “exotic” operations
$x\oplus y=xy$ (usual multiplication)
and
$k\odot x=x^k$ .
a. Show that ${ℝ}^{+}$ with these operations is a linear space; find a basis of this space.
b. Show that $T\left(x\right)=\ln \left(x\right)$ is a linear transformation from ${ℝ}^{+}$ to $ℝ$ , where $ℝ$ is endowed with the ordinary operations. Is T an isomorphism?