These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Angst? What Angst?: The protagonists and other former members of the Debauchery Tea Party take their unexpected imprisonment in a video game surprisingly well, and complain that most of the other players are getting too worked up over it. There are many players who angst about it, but those players are not the main characters. It helps that their actual lives are not directly threatened; they respawn when killed. The players are more concerned about incertitude and social troubles than surviving.
The animation studio switch from Satelight (season one) to Studio DEEN (season two) split the fan-base as to whether it was an overall upgrade or downgrade. Or, you may even be in the third camp stating that nothing feels different.
Akatsuki's side-story in season two where she goes after the Christmas killer is very hit-and-miss amongst the fans. Some found that the arc didn't need to drag on for four episodes, and were getting impatient, because what they truly wanted to see was the action going on in Shiroe's raid on Elder Tail's gold stash. Others were fine with it, stating that this is what Log Horizon is best known for building up the problem before finally dealing with it, and eventually got to what everyone was begging for; an action-packed final episode of the arc dealing with the murderer. Akatsuki's Character Development seen throughout this arc is also a huge plus for some people as it was something that was constantly asked for all throughout season one due to her Ensemble Darkhorse status.
Nureha, one of her rivals for Shiroe's affection, also has a significant following among Japanese readers, though for an entirelydifferent reason. She's the one with the Courtesan (娼姫, literally "whore princess") subclass, by the way.
After her debut in the anime, Serara is also highly popular among viewers.
Isaac's popularity exploded, at least amongst the Japanese fandom, after the release of the Black Knight vs. Berserker Drama CD in volume 7's special edition.
Game Breaker: Basically speaking, Shiroe does this on a regular basis. His first unmistakable demonstration is buying the guildhall of Akihabara. This is only the first of many such feats and he performs many others... say for example, unlocking how to promote Landers into Adventurers. These things have profound implications that can affect tens of thousands... if not the entire world.
Hilarious in Hindsight: Much of the beginning of Season 2 involves Shiroe trying to figure out how to generate enough funds to pay for the Round Table's upkeep costs for maintaining Akihabara's infrastructure. He then comes across an idea to exploit a hole in the game's logic: mainly, if monsters drop gold, where does the gold from? His investigation takes him to an underground raid zone where it is rumored that it holds an endless supply of gold. So basically, Shiroe is looking for the fabled Loot Cave.
Hollywood Pudgy: Averted in Kyouko's case, as the way she's drawn she actually does look pudgy, especially when compared to the rest of her guildmates, who are almost universally slim.
Kanami. Readers of her spinoff joke that the only thing missing is for Landers to worship the land she walks on.
Soujirou manages to fend off a Royal Guard, even intercepting it and blocking its blows multiple times, while it's supposed to be impossible to do even once. Bundled with his harem guild, well-liked personality, his rare subclass, and renowned sword-fighting capabilities, people tend to call him Kirito 2.0. More people dislike him after the Black Knight vs. Berserker Drama CD, wherein he effortlessly intervenes in a duel between the two best Guardians on the Yamato server just as they were about to settle it, simply to use the space to hang out with his guild. Unlike the other guild leaders like Shiroe, Krusty, and even Isaac, Soujiro doesn't really come up with too many strategies on his own for facing enemies, usually just relying on sheer skill and luck. His only two deaths are both against Royal Guard, who are supposed to be able to one-shot Adventurers, and the second time is against a Royal Guard using a Raid Boss-possessed weapon that boosted the Guard's power even more. Even then, Souji is able to parry and block his attacks for quite a while.
Mary Tzu: Shiroe's plans rarely fail and often lead to something when they do. Shiroe regularly executes or theorizes exploits that are game breaking, which others find astonishing because they couldn't imagine it themselves (despite how obvious it would be for real world MMO veterans). He is a recognized genius because of these factors, and people want information on whatever he's scheming on the sole merit that it's a Shiroe Plan. He's so good, people in-story wish they had a fraction of the intellect he does. From a Military Science prospective he's demonstrated the genius of Sun Tzu (a battle can be won before it's even fought through planning), Machiavelli (duplicity and subterfuge is justified to achieve a political goal), and Clausewitz (the threat of force is a form of politics)... just to name three major figures out of the dozens he emulates. Economic Science too! It's far more detailed in the light novel, but only touched upon the anime to save time, that he's demonstrated insight with John Keynes (father of modern capitalism) and Milton Friedman (a critic of Keynesian Economics with alternative insight) to avoid most obvious pitfalls. Some of the Trade Guild Leaders are amazed at it... despite the fact they should be more knowledgeable about those concepts than Shiroe. It somewhat gets lessened due the MMO world's Hollywood Economics and Cast From Money being different from 20th and 21st century Earth, but the basic principals are sound. As the story progresses Shiroe will also gain the power of Combat Clairvoyance, which obviously needs little explaination to describe how more dangerous it makes him.
Shipping: Mostly avoided. Much like in his previous work, Maoyuu, Mamare prefers One True Pairing, as evidenced by the front and back covers of book 5 (which showcases the more obvious pairings). However, he has acknowledged that he conceptualized three possible love interests for Shiroe, which obviously has gotten the shipping-inclined among the fandom very interested on who's the "best" girl for him.
They Changed It, Now It Sucks: Some fans have complained about Shiroe taking significant damage in their first battle, instead of only trivial amounts in the novel... even though his dwindling health is only used as a cliffhanger for the In Medias Res opening scene, and isn't brought up when the full battle is shown.
Some of the censorship in the anime, such as the lack of blood, cutting off Naotsugu's perverted lines (courtesy of Akatsuki's flying knees) as a running gag, and hiding Nureha's backstory have caused complaints about the show. However, this is Justified since the show airs on NHK, which is a publicly funded network that caters to children.
Readers of Sword Art Online have claimed that a major plot twist of Log Horizon ( the NPC population is sentient) was also in Sword Art Online. It's really a Cryptic Background Reference, but cited on a few occassions (once in the anime) as a reason why Kirito dislikes having innocent NPC's killed during Boss Raids. Because SAO is mostly a single-person focused story, the matter isn't explored beyond Yui's involvement (and Fina to a lesser degree). Again though: TRON and Neuromancer did it first... so it's silly to nitpick.
The premise of an MMO being real, with humans having a profound impact on it (and vice versa) was the ambiguous ending to the game Star Ocean: Till the End of Time.