Where the city council has designated an administrative officer to carry out the duties of the findings and hearings in sections 4-4-1 through 4-4-7 of this chapter, the property owner shall have a right of appeal to the city council from any order of the administrative officer. Such appeal shall be taken by filing written notice of appeal with the city clerk within ten (10) days after the administrative hearing. (1990 Code § 8-108)