I would like to find (all) associative binary operations of the form
$$u_{1}*u_{2}=\ln{\left[e^{u_{1}}+e^{u_{2}}\right]}+Q\left(u_{2}-u_{1}\right),$$ where $Q$ is an arbitrary function.
My effort:
beside the trivial $Q\equiv0$ that provides the associativity of $*$, function $$Q\left(u_{2}-u_{1}\right)=-\ln\left[e^{u_1}+e^{u_2} \right]-\ln\left[e^{-u_1}+e^{-u_2} \right]$$ can be written as $$Q\left(u_{2}-u_{1}\right)=-\ln\left[2+e^{-(u_2-u_1)}+e^{(u_2-u_1)}\right],$$ so that $$u_{1}*u_{2}=u_1+u_2-\ln{\left[e^{u_{1}}+e^{u_{2}}\right]}$$ is an associative binary operation.
By random trial and error, I have checked many other functions $Q$ but failed to obtain associativity of $*$. Is there a systematic way to determine what $Q$ provide for associativity of $*$?