Assume that ($\oplus$, $\otimes$) is a semiring over the positive reals.

If $\otimes$ is +, does that imply that $\oplus$ is logsumexp, thereby making ($\oplus$, $\otimes$) the log semiring?

Clearly $\oplus$ = logsumexp is a solution but I would like to prove or disprove its uniqueness.