local authority vs office

local authority and office both are nouns.

Word	Noun	Adjective	Verb	Adverb
local authority	Yes	No	No	No
office	Yes	No	No	No

As nouns, office is a hypernym of local authority; that is, office is a word with a broader meaning than local authority:

Other hypernyms of local authority include agency, authority, bureau, federal agency, government agency.

local authority (noun)	office (noun)
an administrative unit of local government	a religious rite or service prescribed by ecclesiastical authorities
	the actions and activities assigned to or required or expected of a person or group
	a job in an organization
	place of business where professional or clerical duties are performed
	professional or clerical workers in an office
	an administrative unit of government
	(of a government or government official) holding an office means being in power