Logan outlined a list of four abusive comments alleged to have been aimed at him during Cove's 1-0 cinch League One victory, with one of them racist.
Writing on his Twitter account, the former Aberdeen full-back headed his post with laughing emojis and added: "When you upset Clyde fans because your mum chose a black man."
The 33-year-old added the hashtags "#Kickitout" and "#BlackandProud".
Clyde were quick to respond following Logan's comments.
A statement from the Cumbernauld side read: "The club has been made aware of an accusation of a racist comment being made during today's cinch League One match at Broadwood.
"Clyde Football Club will not tolerate or condone racist, sectarian, sexual or bigoted harassment or other discriminatory behaviour, whether physical or verbal, and will ensure that such behaviour is met with appropriate action in whatever context it occurs.
"The club has tonight instigated an investigation into this matter."
Police are also investigating allegations that racist abuse was directed at former England defender Ashley Cole at Swindon's FA Cup clash with Manchester City on Friday night.
Swindon have apologised to Cole, who was pitchside as part of ITV's presentation team for Town's 4-1 third-round defeat at the County Ground.
Supt Phil Staynings from Wiltshire Police said in a statement: "We are aware that one person was subject to racial abuse at the County Ground following the Swindon Town v Manchester City game last night.
"We have already commenced an investigation, obtaining statements and securing CCTV. Further enquiries in relation to this incident are ongoing.
"Behaviour of this nature is completely unacceptable and will not be tolerated. Anyone with information should call 101 and quote crime reference number 54220002136."
Swindon issued a statement condemning the actions after revealing they received reports of racial abuse towards Cole.
Owner Clem Morfuni said in the statement: "We are absolutely saddened and deeply disgusted by this and we cannot stress strongly enough that racial abuse of any kind will not be tolerated by Swindon Town Football Club.
"Racism has no place in the world and it is incredibly heart-breaking that this still has a place in our game.
"We are currently working with Wiltshire Police on gathering evidence but those responsible will be punished accordingly.
"On behalf of everyone at Swindon Town Football Club, we send out our heartfelt apologies to Ashley Cole and it is with our deepest regret that you had to experience that last night.
"Those few individuals do not represent Swindon Town and will not be welcome at The County Ground."