Before Logan County was formed it was Sangamon County. Except for land records for information on Logan County between the first settlement in 1819 until 1839 you must look in Sangamon County records. Most Sangamon County records for that period are housed at IRAD including Board of Supervisors' records and files, various Circuit Court files, election records, marriage records, probate records.

Do not rule out any records. From the election records I found the following list of voters for Union Precinct [a precinct name I was not familiar with] on August 2, 1824:

Charles Counsel
Job Burden
Jacob Bryan
Mathew Rogers
Oliver W. Kellogg
Levi Cantrill
Archibald Constant
William Carpenter
Andrew Cline
Samuel Rhodes
Nathan Hussey
Joshua Sayles
Humphrey Taylor
Thomas Brock
John Orendorff
George Power
Robert Evans
Joseph Beedle
William Brock
Gardner Randolph
Joseph W. Center
Daniel Harper
Evans Brittin
Isace Morgan
Robert McClelland
Jonathan Hodge
Jeharias Lankston
Samuel Carman
John Hendricks
James W. Estill
Thomas Strawbridge
Richard Wright
William Estill
John Stevenson
Hezekiah Davis
Edward Davis
William Johnson
Jacob Dean
William Willcoxan
Nathan Kenneda
Hezekiah Briggs
Jesse Sutton
John Renhart
Peyton Mitchel
Meredith Cooper
Thomas Briggs
James Stewart
Hiram Robins
Bostin Finder
Garret Elkins
John Sutton
James Strawbridge
Stephen Phelps
Ambrose Cooper
Thomas Branson
Martin Porter
Christopher Orendorff
Samuel Carpenter
James Williams
Wyatt Cantrall
John Dorson
John Clin
Cornelius Vandeventer
John W. Taylor
William Strawbridge
Andrew Branson
Eli Branson
Jojin Branson, Jr.
Isiah Stillman
William A. Hodge
Joseph Woods
Philip Smith
Washington Turley
John Woods
David L. Taylor
Stephen Stillman
George Borders
Charles Hayes
Robert Downing
John Branson
James W. Chapman
Alexis Phelps
Henry Cooper
Robert Buckles
Enoch Smith
Benjamin Briggs
Jeremiah Birks
James Turley
Alexander McNaghtin
Abraham Musick
John Simpson
William Prockter
Charles Turley
David Riddle
Levi Millard
James Garvin
Janes Sayles
James Morrow
John Brook
James Latham
David Hughs
Nicholas Moore
Joshua Dean
Myron Phelps
Landa Hurst
Thomas Prim
James C Stevenson
James Stringfield
Henry Holland
Abraham Hathaway
John Dixson
John Porter
Thomas Constant
Peter Vanburgen
Willima Lofton
Ezra Knapp
John Stevenson
Charles Smith
David England
Samuel McClure
John Prim
Robert McClure
John Buckles
John Strode

I don't know where Union Precinct was but I recognize a lot of Logan County names including ancestors and collaterals who were living along Salt Creek at that time as well as people living in the south end of the county in the Lake Ford area. Presumably at that time polling places were few and far between.