I need to set up in Maple the following binary operation $*$: $$\phi(u_1*u_2)=\ln{\left[e^{\phi(u_1)}+e^{\phi(u_2)}\right]}+Q\left[\phi(u_2)-\phi(u_1)\right],$$
where $Q$ can be chosen to be a specific function (linear, polynomial, exponential or any other elementary function) of its input, while $\phi$ is kept the way it is (just "some function of $u$").
The main purpose of this Maple exercise is to check associativity of operation $*$ for various choices of $Q$.
Without $\phi$ (or $\phi$ being an identity map), I think I could check the associativity of $*$ with basic Maple syntax. But keeping $\phi$ unknown makes it confusing. Any advice on how to introduce this operation in Maple?