Political “logrolling” (or “log-rolling”) is a tit-for-tat exchange. In early America, logs had to be rolled to clear land and one person often needed the help of others. When one person rolled logs for another, it was understood that the favor would be returned. In political “logrolling,” as explained in the April 11, 1823 Daily National Intelligencer, the reasoning is: “Do you vote for my measure, and I’ll vote for yours.”
 
The political version of “logrolling” is cited in print from a least 1820. Literary “logrolling”—authors endorsing each other’s works—is cited in print from at least the 1840s. The New York City satirical magazine Spy had a “literary logrolling” feature in the 1980s-1990s, detailing shameless book-jacket co-endorsements.
   
     
Wikipedia: Logrolling
Logrolling is the trading of favors or quid pro quo, such as vote trading by legislative members to obtain passage of actions of interest to each legislative member. It is also the “cross quoting” of papers by academics in order to drive up reference counts. The Nuttall Encyclopedia describes log-rolling as “mutual praise by authors of each other’s work.” American frontiersman Davy Crockett was one of the first to apply the term to legislation:
 
The first known use of the term was by Congressman Davy Crockett, who said on the floor (of the U.S. House of Representatives) in 1835, “my people don’t like me to log-roll in their business, and vote away pre-emption rights to fellows in other states that never kindle a fire on their own land.”
 
The widest accepted origin is the old custom of neighbors assisting each other with the moving of logs. If two neighbors had cut a lot of timber which needed to be moved, it made more sense for them to work together to roll the logs. In this way, it is similar to a barn-raising where a neighbor comes and helps build your barn and then you go and help build his.
 
C-SPAN Congressional Glossary
Logrolling is the term used for an informal pact between members to vote for each other’s priorities.
     
Merriam-Webster Dictionary
Main Entry: log·roll·ing
Pronunciation: \-ˌrō-liŋ\
Function: noun
Date: 1812
1 [from a former American custom of neighbors assisting one another in rolling logs into a pile for burning] : the exchanging of assistance or favors ; specifically : the trading of votes by legislators to secure favorable action on projects of interest to each one
2: the rolling of logs in water by treading ; also : a sport in which contestants treading logs try to dislodge one another
   
(Historical Dictionary of American Slang)
logroll v. Esp. Pol. to practice favoritism or cronyism; (specif.) to use political influence or connections; accomplish or obtain (something) through such influence. Now colloq. Hence logroller, n.
1812 (Cited in W10)
1823 in OED2: “That sort of “management,” now rather more fashionable, and known by the dignified appellation of “log-rolling”—that is, a buying and selling of votes.
   
(Oxford English Dictionary)
log-rolling
colloq. (orig. U.S.). 
a. Combination for mutual assistance in political or other action. Also attrib. or as adj.
Suggested by the proverbial phrase ‘You roll my log and I’ll roll yours’.
1823 Niles’ Weekly Reg. 7 June 210/1 That sort of ‘management’, now rather more fashionable, and known by the dignified appellation of ‘log-rolling’—that is, a buying and selling of votes.
1838 J. A. QUITMAN Let. 13 Dec. in J. F. H. Claiborne Life & Corr. J. A. Quitman (1860) I. 165 Tending to promote combinations and log-rolling schemes.
1841-4 EMERSON Ess., Poet Wks. (Bohn.) I. 169 Our log-rolling, our stumps and their politics..are yet unsung.
1860 S. MORDECAI Virginia xxx. 303 But the log-rolling system of Virginia has diverted her energies from the completion of any one useful work.
1869 Atlantic Monthly Sept. 365/2 The log-rolling lobby generally exerted their powers upon objects which possessed a public character.
1879 Times 19 June, The bribe was political preferment, or ‘log-rolling’—that is, help in passing other Bills.
1888 BRYCE Amer. Commw. I. I. xv. 213 Corruption..appears chiefly in the milder form of reciprocal jobbing or (as it is called) ‘log-rolling’.
 
b. Mutual puffing in literary publications.
[1845 in Longm. Mag. (1900) Feb. 375 Somewhere in this book of Letters occurs, about 1845, the phrase ‘literary log-rolling’, the earliest instance which one has met.]
1888 J. PAYN in Illustr. Lond. News 7 Jan. 2 To have an eye to its [the book’s] merits rather than to its defects, is obviously log-rolling.
 
log-roll, v.
[Back-formation from LOG-ROLLING.]
a. trans. To procure the passing of (a bill) by log-rolling.  b. To approach (a politician) with the view of getting his political co-operation.  c. intr. To engage in log-rolling.
1835 D. CROCKETT Tour 120 My people don’t like me to log-roll in their business, and vote away pre-emption rights to fellows in other states, that never kindle a fire on their lands.
1837 H. MARTINEAU Soc. Amer. II. 279 The method of ‘log rolling’ bills through the legislature.
1865 Daily Tel. 14 Apr., The leading politicians who..log-roll the railway bills.
1876 LOWELL Among my Bks. Ser. II. 98 In the Greek epic, the gods..lobby and log-roll for their candidates.
1879 Times 19 June, To log-roll with everybody who was willing to work with him
 
23 March 1784, Massachusetts Gazette, pg. 3:
On Tuesday the 9th inst. as Mr. EBENZER BROWN, of Long-Meadow, was loading a large log on his sled, one of the stakes gave way, by which accident the log rolling off caught his feet,...
   
3 March 1821, Batimore (MD) Patriot, “Bankrupt Bill,” pg. 2:
A member of congress, who is opposed to the bill, informs us, that nothing but the want of time prevented its passage this session, there being a decided majority in its favor. He says there was an understanding, a “kind of log rolling” between the friends of the bankrupt and land bill.     

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Letters From America
By James Flint
Edinburgh: W. & C. Tait
1822
Pg. 188:
Jeffersonville, (Indiana,) March 10, 1820.
Pg. 189:
Much of the business (it is said) is privately arranged, before the questions are discussed in the house. Combinations are formed for effecting particular purposes. These are called log rolling; a very significant metaphor, borrowed from the practice of several farmers uniting in rolling together large timber to be burnt. A number of bills are frequently conjoined by their movers, so that a member who takes a deep interest in one must vote for all of tehm, to obtain the suffrage of the separate partizans. The member who deserts from the cabal might be leaving his own motion without any other supporter but himself. An enlightened gentleman told me, that he was induced to vote for the ridiculous law of this State regarding intercourse between whiite and coloured people, in consequence of its being previously conjoined with other bills. 

10 July 1822, Sandusky (OH) Clarion, pg. 2, col. 2:
The Ohio Legislature—While log rolling is the order of the day, what good can be expected? May good come out of evil.
 
11 April 1823, Daily National Intelligencer (Washington, DC), pg. 2:
LOG ROLLING.
This term (log-rolling) is used by a correspondent in a letter published on Saturday, but as it is ten to one if one out of ten of our readers understand its import, an explanation is proper.
 
The difficulty of comprehending this term arises not from the words which enter into its composition being foreign or uncommon. On the contrary, they are both genuine English words, and their combination is in strict accordance with the analogy of the English language. In the canting sense, however, in which they are applied to legislative proceedings, many persons may, unless some assistance be given, experience the same difficulty in fixing their meaning, that we ourselves did when we first heard the term used.
 
The metaphor, (for such it in fact is, however homely it may appear to persons of classical taste,) is, as we are informed, borrowed from the customs of the backwoodsmen. “Do you help me roll my log, and I’ll help you roll yours.” This is the literal sense. “Do you vote for my measure, and I’ll vote for yours.” This is the metaphorical. This is legislative log-rolling; and in this manner are various acts, good, bad, and indifferent, all rolled together into the statute book.
 
A long dissertation is not required to shew the pernicious tendency of this system. Its evil consequences must be self evident to even the meanest capacity. It supposes questions of public policy to be decided not according to their own merits, but according to the merits of other questions which have no necessary connection with those first mentioned. It supposes a system of bargain and sale established among the representatives of the people, who should know no motive but the public good, and “who should never do evil that good may come.’—Phil.  Gaz.
 
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27 June 1829, Niles’ Weekly Register, pg. 284, col. 1:
...while those of Philadelphia loudly condemn, and we think justly, what is called the “log-rolling system”—or bargaining in the legislature, for the making of one canal by agreeing to make another; thus beginning many great works at one time and wasting the public means, without completing any thing.
 
11 April 1884, New York (NY) Times, “THE PUBLIC BUILDING RAID; THE HOUSE PORK BARREL STILL YIELDING PRIZES,” pg. 1, col. 4:
Although “pork” may be alluded to in the House, and no member is overstepping the rules by using that vulgar word, “log rolling” is a term that is permitted only to newspapers and the plain-speaking outside public.