A person who becomes owner of a dog that is 4 or more months old and that is not already licensed shall apply for a license within 30 days. A person who owns a dog that will become 4 months old and that is not already licensed shall apply for a license within 30 days after the dog becomes 4 months old. In a county in which section 6(2) or section 6(3)(a) applies, if a person applies for an annual license under this subsection after July 10 of a calendar year, the license fee shall be 1/2 the fee provided for under section 6.