I was looking at the function $xy=1$ and realized eventually, that exponentiating all points on this curve yields points on the curve $\ln(x)\ln(y)=1.$
For example: $(1,1)\mapsto(e^1,e^1)$ and $(2,1/2)\mapsto(e^2,e^{1/2}).$
Is this a correct way to write the map between the points from one function to the other? $(x,1/x)\mapsto(e^x,e^{1/x}).$
Should I think of $\ln(x)\ln(y)=1$ as the exponential map of the curve $xy=1?$
Upon decoding your cryptic message came forth these random thoughts.
It is not the exponential map.
It is not the logarithmic map.
The logarithmic map is ln x + ln y = 0.
Whereupon xy = 1.
If this is not correct, then forsooth be thy cypher?