I have exactly the same problem.
You don't need the solder-tooling jar within the Arquillian test itself.
solder-tooling needs to be on the classpath of the project that has your test code, which would include the Annotated interface, if you are using Type Safe Logging.
What's the fully qualified class of the logger you're trying to inject?
I think I should RTFM myself, after all you don't need to inject the Logger
Invoking the logger should be as follows:
Logger log = Logger.getLogger(WhateverClass.class);