If sounds the same with and without power, then the ESL energizing circuit is not coming on or is busted.
This could be due to two (maybe more) things:
1- The 'Music Sense' circuit has failed and is not triggering the power to the rest of the curcuitry
2- Some other component in the ESL diaphragm energizing circuit is bad
3- (see I did say more) the ESL Step-up got fried by a big surge from an amp.
Even though I owned a Logos for many years, I never opened it up, but from what I've read here, it's complex. Do a search on the site.
Once in the unit, measure voltages from the music sense circuit to make sure that is triggering.
Also, if there any fuses in there, test and replace them.
You can test the step-up transformer just like any other transformer to make sure it's not shorted or cut.
See this thread for more info on working on a logos:
Putting a new membrame in the Logos
Suggest you try contacting the author, but their last post here was in 2013.
BTW- The Logos is one of the few models ML no longer provides parts for. So it's DIY for any and all fixes.