The 24-year-old was handed the suspension after he was found guilty of using "offensive, insulting and abusive language of a racist nature".
After the ruling was announced, a Celtic spokesperson told the club's official website: "Racism has no place in football and as a club for all people, Celtic absolutely abhors racism of any kind.
"This was a very unfortunate case, but the club has accepted Aleksandar's explanation that he did not say the words that were alleged to have been said and that he is not a racist.
"We are, therefore, very disappointed by the outcome today and can confirm that Aleksandar will be appealing this decision."
Tonev, who is on loan at the Scottish club from Aston Villa, is believed to have used the language in an incident with Logan during Celtic's 2-1 win over Aberdeen last month.