Using Log Horizon for a company name would be... a little weird. I am I'm a fan of the series but the entire point of the guild Log Horizon was to get other guilds to work together and it only has 6 members in the series.
Log Horizon actually got me to get back into FFXIV after watching it. but I still wouldn't name a guild after it, just doesn't fit with its meaning. Why not name it after one of the other guilds of Ahkibari? Like "The Silver Sword".
Eventually, I plan to make a Free company. but only after I become a master botanist, master carpenter and can afford a large plot of land in character housing.