The first county seat was at Postville. The second county seat was at Mt. Pulaski. In 1856 the county seat was moved to Lincoln, a newly formed town named after Abraham Lincoln, the only town to be named for him before he became President. 
Logan County was originally laid out in 13 Precincts: Atlanta, Bowles, Broadwell, Elkhart, Eminence, Lake Fork, Lincoln, Madison, Middletown, Mt. Pulaski, Prairie Creek, Salt Creek, Sugar Creek. We do not know the boundaries of the original precincts or where each one was. Townships were formed along the standard lines under the Northwest Ordinance.
Around 1867 17 Townships were formed from the Precincts: Aetna, Atlanta, Broadwell, Chester, Corwin, East Lincoln, Elkhart, Eminence, Hurlbut, Laenna, Lake Fork, Mt. Pulaski, Oran, Orvil, Prairie Creek, Sheridan, West Lincoln.
Place names - not necessarily populated places - for which at least one reference has been found are; Aetna, Albany, Allenville, Altic, Atlanta, Athol, Atlanta, Atteberry, Beason, Bell, Big Prairie, Bloomingdale, Boren, Broadwell, Burton View, Camden, Camp Latham, Chester, Chestervale, Chestnut, Clay, Cornland, Corwin, Crohurst, CroHurst Station, East Lincoln, Elk Hart City, Elk Heart City, Elkhart, Elkhart City, Emden, Eminence, Evans, Fogarty, Fulscher, Gale, Georgetown, Gilchrist, Gillett, Gold Springs, Griggs, Hamilton, Harness, Hartsburg, Hartsburgh, Hartwig, Hurlbut, Hurricane Point, Irish Grove, Johnsons, Johnsons Siding, Kickapoo, Kickapoo Siding, Kings, Kruger, Laenna, Lake Fork, Latham, Lawndale, Layman, Lee, Leesburgh, Lincoln, Lincoln Hill, Long Point, Lucas, Mackey, Madison, Martin, Middletown, Mt. Fulcher, Mt. Hope, Mt. Joy, Mt. Pulaski, Music, Musick's Ferry, Narita, New Castle, New Holland, North Lincoln, Oran, Orndorff, Orvil, P & N Junction, Peifer, Peoria District Junction, Perry, Postville, Prairie Creek, Pulaski, Quisenberry, Rhody, Richmond Grove, Salt Creek, San Jose, Scroggin, Sheridan, Skelton, Union, Utility, Warick, West Lincoln, Wilmert, Woodlawn, Woolf, Wyatt, Xenia, Yankeetown. 
The Logan County Illinois Genealogical and Historical Society  in Lincoln and the Mt. Pulaski Township Historical Society in Mt. Pulaski are good places to start your in person search. Both are staffed by knowledgable people. When traveling from a distance it is advisable to call ahead.
Your online search should begin at the Logan County ILGenWeb site . The main page has links to the state's resources and county specific resources as well as resources on the site including listings for many cemeteries, 1840 and 1850 census indexes, links to free views of the actual 1840 and 1850 census as well as the 1830 Sangamon County census which included Logan County, transcriptions of the 1855 Illinois census and 1862 military census, listings of residents from old histories and much more.