Maintenance Controller reported confusion and interruptions caused a procedure to be left incomplete. This resulted in an aircraft departing with an open log page. The flight crew sent the required response after take off.

Narrative

After push back off of Gate XX; The flight crew on Aircraft X called the hangar controller radio stating they had an Anti Ice Standby Pitot AOA message. I reviewed the history of the aircraft and proceeded to have the crew do a circuit breaker reset for the Standby AOA. The fault cleared and the crew was informed to send an ACARS code when able; to log the reset into the history. At this time the crew did not mention they had made a log for this and would need a new Maintenance Release. When I previously had checked; this aircraft had a valid Maintenance Release. Maintenance later called to notify me that the aircraft departed with an open log and did not have a valid Maintenance Release due to this. The flight crew did send the code after take-off however (ELB log XXXXXXX) at XA:47 local; they took off at XA:14.During the time I was working with the flight crew; we were on the tail end of a busy bank of flights and also several arrivals calling inbound logs. My time was not uninterrupted as I was multi-tasking with other tasks simultaneously. I have never had an aircraft leave without a valid Maintenance Release before this event and believe there was a failure on my part to re-check the ELB and also on the part of the flight crew to not communicate the log was open and they needed a new Maintenance Release. Working solo in this position puts far too much pressure on one individual and creates situations where you are distracted in many directions (phone calls; radio calls; operations center chat updates; vendors; people in the operations center approaching you when you are dealing with aircraft on the radio) all can lead to a situation like this where you lose full concentration and then move onto the next task without the time to properly go over the current event and make sure every step of our process has been followed.Always having two or more hangar controllers working per shift so one is able to work without being distracted when working technical issues on aircraft off the gate. The number of distractions can be extremely high and takes away our ability to fully concentrate on one task at a time. Double checking the ELB prior to an aircraft departing after pushback.