I searched the Irish Family History Foundation's database. It contains 12,000,000 surviving rcds for abt 65% of Irish counties. B4 1840 rcds are sparse, B4 1800 rare. My search is limited to this database.
For Elizabeth Logan M 1881 in Co. Leitrim, I found 1 M rcd. This is likely for your ancestor. A view of the detailed rcd SHOULD show their parents' names, residences, etc.
For Elizabeth Logan B 1858 +/- 4 yrs in all Ireland, I found 4 B rcds: 1856 Roscommon, 1860 Antrim, 1861 Longford, and 1862 Antrim. None of these listed Joseph as the father's forename.
For Joseph Logan M 1853 +/- 4 yrs in all Ireland, I found 2 M rcds: 1849 Antrim and 1850 Down. Modify the search range for additional possibilities.
For possible Logan brothers with a father named Joseph, I found the following: John Joseph B 1838 Dublin, Joseph B 1840 Cavan, Aloysius B 1845 Dublin, Aloysisus B 1845 Dublin.
Google the Foundation's name to lead you to this database. Pls post your search results to my genforum msg.