The contract will cover the maintenance of all the fire extinguishers throughout the building to ensure all occupants are safe and that the Chamber is fully compliant with current regulations.
The Chamber moved to its new premises in Queen Street from its original home in York Place just before Christmas 2013.
Logic will carry out annual maintenance work until 2019, which will see engineers inspect for damage, ensure they are working within require parameters and are in their correct location.
Gary Dale, operations director at Logic Fire & Security, said: "While our client base stretches far and wide across the UK, it's always pleasing to win a contract on our doorstep.
"Through our membership of the Chamber we knew they would be required to assess their fire safety commitments in their new office, so it was a simple case of us being in the right place at the right time.
"Our impressive portfolio of clients coupled with, in this instance, our track record with maintenance all helped us to secure the contract."
Logic Fire & Security was founded in 2001 as a specialist turnkey fire alarm company, and has since expanded to offer intruder alarms, CCTV and access control solutions.
Its Yorkshire clients include the University of Leeds, Leeds City Museum and Shoreline Housing Partnership;