mp.mathematical physics - Asymptotic Matching of an logarithmic Outer solution to an exponential growing inner solution

Your question is a bit vague. Indeed, the standard procedure would involve rejecting the singular solutions and then using a combination of inner and outer expansions to satisfy the boundary conditions. There are various specific methods of achieving the goal (Vishik-Lyusternik and the matched asymptotic expansions are the most popular), but typically, one or several boundary conditions are only satisfied asymptotically (i.e. with an error vanishing as the small parameter tends to its limit, with the error often being exponentially small). Therefore, if some (or all) of your boundary conditions are satisfied only approximately (i.e. only in the limit of vanishing small parameter), this may in principle be the "feature" of the method that you are using. Otherwise, if the boundary conditions cannot be satisfied in this sense, it usually signals of the presence of yet another boundary layer which needs to be accounted for.

Kevorkian and Cole is an excellent source; you may also find helpful Van Dyke's "Perturbation methods in fluid mechanics" (a bit terse), Nayfeh's "Perturbation methods" (textbook) and de Jager and Furu's "The theory of singular perturbations" (good alternative).