The Scottish universities, generally, would record the name of a student's father, so in order to find out anything about your Alexander Logan, you need to tell us which university he graduated from.
In the 1770s there were just seven universities in Britain - five* in Scotland and two in England.
The University of Glasgow used to have a useful list of its graduates online, but it seems to have disappeared.
University of Aberdeen graduates are listed in a set of books which are readily available in libraries, and have been partly digitised.
The Universities of St Andrews and Edinburgh have not, as far as I know, published similar lists.
Your best plan would be to contact the Archives of whichever university Alexander Logan graduated from.
*University of St Andrews, founded 1411; University of Glasgow, founded 1451; University of Aberdeen, founded 1495; University of Edinburgh, founded 1583; Marischal College, founded 1593 and amalgamated with the University of Aberdeen in 1860.