Auburn is a home rule-class city in Logan County, Kentucky, in the United States. The population was 1,340 at the 2010 census. Originally calleā¦Auburn is a home rule-class city in Logan County, Kentucky, in the United States. The population was 1,340 at the 2010 census. Originally called Federal Grove, the present name dates from the 1860s; it was named after Auburn, New York, the previous residence of several early settlers. It was originally incorporated by the state assembly in 1865, and reincorporated in 1878.