Does ln((e^u)+(e^v))=u+v or uv?

Question

Thanks for clarifying!

Answer 1

neither, you are confused in the basic law:
ln(xy) = lnx + lny ≠ ln(x+y)

testing your result,
let v = 2, u = 3
then ln(e^3 + e^2)
= ln(27.47745..)
= appr 3.31
≠ 3+2 nor 2*3

Answer 2

What would ln(e^u+e^v) equal then?