There is a 1870 census record for a Joseph Young listed in Boon, Scott County, Arkansas, P.O. Booneville. (Booneville is also in Logan County, right?):

YOUNG, Joseph, age 45, born Missouri, farmer, also listed in the household are: Almedia, age 25, born Missouri, keeping house, Prissila, age 11, William, age 10 and Talletha A., age 2 mos. (children were born in Arkansas).

As Bob said, more information of possible names related to the family would be helpful in locating her family.