Is it possible to define an inner product in this vector space to make it an inner product space?

Question

This is for curiousity; whether or not it is possible and if so, how/what would be the inner product?
Vector space is defined by $(\mathbb{R}^{+},\oplus,\otimes,\mathbb{R})$
where $x\oplus y = x \times y$ and
$x \otimes y = y^x$.

Answer 1

Hint

Given that your vector space is isomorphic to $(\mathbb R, +,\times, \mathbb R$) with standard scalar multiplication and addition, and knowing that the isomorphism is $$x\mapsto e^x$$

you can start from here.

You know that in the standard vector space, $\langle x, y\rangle=x\cdot y$ is an inner product. So, the idea is that you take $x,y$ in your vector space, map them to the standard vector space, and calculate their inner product there.