axe (v.) Look up axe at
1670s, "to shape or cut with an axe," from axe (n.). Meaning "to remove, severely reduce," usually figurative, recorded by 1922. Related: Axed; axing.
axe (n.) Look up axe at
Old English æces (Northumbrian acas) "axe, pickaxe, hatchet," later æx, from Proto-Germanic *akusjo (source also of Old Saxon accus, Old Norse ex, Old Frisian axe, German Axt, Gothic aqizi), from PIE *agw(e)si- (source also of Greek axine, Latin ascia).
The spelling ax is better on every ground, of etymology, phonology, and analogy, than axe, which became prevalent during the 19th century; but it is now disused in Britain. [OED]

The spelling ax, though "better on every ground, of etymology, phonology, & analogy" (OED), is so strange to 20th-c. eyes that it suggests pedantry & is unlikely to be restored. [Fowler]
Meaning "musical instrument" is 1955, originally jazz slang for the saxophone; rock slang for "guitar" dates to 1967. The axe in figurative sense of cutting of anything (expenses, workers, etc.), especially as a cost-saving measure, is from 1922, probably from the notion of the headman's literal axe (itself attested from mid-15c.). To have an axe to grind is from an 1815 essay by U.S. editor and politician Charles Miner (1780-1865) in which a man flatters a boy and gets him to do the chore of axe-grinding for him, then leaves without offering thanks or recompense. Misattributed to Benjamin Franklin in Weekley, OED print edition, and many other sources.