Is $T$ a nonlinear map?

Question

Define $T:\Bbb R^2\to \Bbb R^2$ with $(x,y)\in\Bbb R^2$ and $(e^x,e^y)\in \Bbb R^2$ s.t. $\forall x,y$ $(x,y)\mapsto (e^x,e^y).$ Define the origin to be $(0,0)$ before the map and after the map. This transformation maps all points in $\Bbb R^2$ to the first quadrant of $\Bbb R^2.$

Q: Is $T$ a nonlinear map? I say this because distances seem to be distorted.

Answer 1

$T(1,0)=(e,1)$ and $T(0,1)=(1,e)$ but $T((1,0)+(0,1))=T(1,1)=(e,e)\neq (e+1,e+1)=T(1,0)+T(0,1)$ so is nonlinear