before vowels log-, word-forming element meaning "speech, word," also "reason," from Greek logos "word, discourse; reason," from PIE root *leg- (1) "to collect, gather," with derivatives meaning "to speak (to 'pick out words')."
word-forming element meaning "battle, war, contest, fighting, warfare," from Latinized form of Greek -makhia, from makhē "a battle, fight," related to makhesthai "to fight." Beekes suspects it is from an isolated root, perhaps Pre-Greek: "In the domain of fighting and battle, old inherited expressions can hardly be expected."
<a href="https://www.etymonline.com/word/logomachy">Etymology of logomachy by etymonline</a>
Harper, D. (n.d.). Etymology of logomachy. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/logomachy