Minister's time invested in incomplete log (6)
I believe the answer is:
rector
'minister's' is the definition.
('rector' can be a synonym of 'minister')
'time invested in incomplete log' is the wordplay.
'time' becomes 't'.
'invested in' is an insertion indicator.
'incomplete' means to remove the last letter.
'log' becomes 'record' (I've seen this before).
'record' with its final letter taken off is 'recor'.
't' placed into 'recor' is 'RECTOR'.
(Other definitions for rector that I've seen before include "Parish priest" , "Vicar" , "Churchman" , "Type of clergyman" , "Member of the clergy" .)