Exercise 11

Consider the vector space $\mathbb{R}^{+}$ where vector addition and scalar multiplication are not the ones inherited from $\mathbb{R}$ but rather are these: $a+b$ is the product of $a$ and $b$, and $r\cdot a$ is the $r$-th power of $a$. (This was shown to be a vector space in an earlier exercise.) Verify that the natural logarithm map $\ln\colon\mathbb{R}^{+}\to\mathbb{R}$ is a homomorphism between these two spaces. Is it an isomorphism?