# Math Help - Prove Linearity of the Map

1. ## Prove Linearity of the Map

Prove the linearity of the map T : (R(>0), +*, **) -->(R, +, *), a --> ln(a) where R carries the standard vector space structure and R(>0) has the vector structure a +* b = ab and a ** b = a^b.

To prove linearity, you have to prove that T(x + y) = T(x) + T(y) and T(cx) = cT(x), but I don't understand what to do with the different vector spaces.

2. (i)

$\forall x,\forall y\in \mathbb{R}^+-\{0\}:\;\;T(x+^*y)=T(xy)=\ln (xy)=\ln x+\ln y=T(x)+T(y)$

Try:

(ii)

$\forall \lambda,\forall x\in \mathbb{R}^+-\{0\}:\;\;T(\lambda^{\;**}x)=\ldots=\lambda T(x)$

Fernando Revilla