Define a coordinate system in $\Bbb R^2$ with an exponential scale on the axes: $\exp(x)-\exp(y).$ Then embed this coordinate system in the first quadrant of $\Bbb R^2$ such that we view the axes now as cartesian $x-y$ coordinates.
You can think of all this as a map $f$ which transforms points in $\Bbb R^2.$
$f:\Bbb R^2\to(0,\infty)\times(0,\infty)$ with $f(x,y)=(e^x,e^y).$
As an example consider a curve $xy=1$ embedded in $\Bbb R^2.$ Change the scale of the axes both to exponential, to obtain $\log(x)\log(y)=1.$ And then let the axes be cartesian $x-y$ coordinates.
Another way to think about it is to transform the points in $\Bbb R^2$ via $f.$ The pre-image space will have the algebraic equation $xy=1$ while the image space will have transformed this algebraic equation to $\log(x)\log(y)=1.$ Note: via the subsitution $u=\log(x)$ and $v=\log(y)$ you can see that $uv=1$ is a hyperbola. This means to transfer from an $\exp(x)-\exp(y)$ coordinate system to a cartesian $x-y$ system you just have to take the logarithm of each ordered pair of points in the space.
Here is a picture depicting the transformation $f.$ I labeled some points $a,b,c,d$ before the transformation and show where they end up after the transformation.
Now there is an exponential-exponential coordinate system of $\Bbb R^2$ embedded within the cartesian first quadrant of $\Bbb R^2$ itself? Can this be right?