Logan was born in 1845, before parents' marriage date. He passed away in 1927.
Sources
1850 U.S. census, population schedule, Floyd County, Kentucky, p. 454, dwelling 585, family 585, Ambrose Arnett; digital images, FamilySearch (https://familysearch.org : accessed 24 August 2017); citing National Archives and Records Administration microfilm M432, roll 200.
1860 U.S. census, population schedule, Licking Station, Magoffin County, Kentucky, p. 467, dwelling 302, family 302, Susan Arnett; digital images, FamilySearch (https://familysearch.org : accessed 24 August 2017); citing National Archives and Records Administration microfilm M653, roll 385.
1870 U.S. census, population schedule, Precinct 4, Magoffin County, Kentucky, p. 6, dwelling 33, family 33, Logan Arnett; digital images, FamilySearch (https://familysearch.org : accessed 25 August 2017); citing National Archives and Records Administration microfilm M593, roll 484.
1880 U.S. census, population schedule, Middlefork Precinct #4, Magoffin County, Kentucky, enumeration district (ED) 63, p. 35, dwelling 212, family 212, Logan Arnett; digital images, FamilySearch (https://familysearch.org : accessed 4 November 2017); citing National Archives and Records Administration microfilm T9, roll 431.
"The Salyer Family: a Genealogical Record of their first 250 years in America." 688 pages with index. by Elizabeth Salyer. Privately published,1982. Reprinted by the Magoffin County (KY) Historical Society. Page A84.
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DNA Connections
It may be possible to confirm family relationships with Logan by comparing test results with other carriers of his Y-chromosome or his mother's mitochondrial DNA.
However, there are no known yDNA or mtDNA test-takers in his direct paternal or maternal line.
It is likely that these autosomal DNA test-takers will share some percentage of DNA with Logan: