# Basis of Vector Space with Non-Standard Definition of Addition and Multiplication

Consider the vector space $$V = \mathbb R^+ \times \mathbb R^+ \times\mathbb R^+$$, where $$\mathbb R^+$$ is the set of all positive real numbers with addition and scalar multiplication defined by

$$(x_1, y_1, z_1) \oplus (x_2, y_2, z_2) = (x_1x_2, y_1y_2, z_1z_2)$$

$$\alpha(x, y, z) = (x^\alpha, y^\alpha, z^\alpha), \alpha\in\mathbb R$$

Find the dimension of the vector space $$V$$ and find a basis of $$V$$ which contains the element $$(e, e, 1)$$

I know it has something to do with exponential or logarithmic function but can't figure it out.

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Indeed, $$V\to\mathbb R^3$$ given by $$(x,y,z)\mapsto (\log x,\log y,\log z)$$ is an isomorphism. It sends $$(e,e,1)$$ to $$(1,1,0)$$, so construct a basis of $$\mathbb R^3$$ containing $$(1,1,0)$$ and send it back through the inverse isomorphism.
Hint: Prove $$V$$ is $$3$$ dimensional, with basis $$\{(e,e,1),(e,1,e),(1,e,e)\}$$. Thus it's isomorphic to $$\mathbb R^3$$.