This is not correct.

It's not the interface that has to be serializable but the bean itself.

So it is the case in the implementation of this interface - which can be either an CDI bean or an EJB. Both implementations must not be serializable and should not too:

It is a Logging-Facade, that's purpose is to log out messages. What should be the purpose of marshaling the "state" of this facade to a client and back again? Neither it has to be passivated and activated it's state - it could be just reinjected by the container itself.

So a 

~~~~
@Inject
transient Log log;
~~~~

is suffiecient.