Logarithmic advice language in P? Is something like DTIME(poly(n))/log(n) in P? Can the log-length advice be somehow hardwired into a DTM for P?
As pointed in the comments, You cannot do this in general,
However, if you're willing to restrict the class and add some some structure you can achive this. For example, you can use the self-reducibility of SAT to remove the advice, i.e if 
$NP\in P/log$ then $NP=P$.

In general, the same is true for complexity classes with self-reducible complete problem, and similar arguments for complexity classes with instance checkers $(EXP,PSPACE,P^{\#P})$ will allow you to remove even larger advices (polynomial) with the difference that now you need to use  probabilistic machines.