Does a database backup reduce size/increase free space of the log file?
Got a SQL database and the log file is massive and keeps growing every day.


Log files growth depends on Recovery Model of a database and transaction log backups.
if database is not critical and data loss is not an issue in any case then it should be in SIMPLE recovery mode. Which allows SQL Server to reuse log space when a transaction is committed and log growth remains in limits. 
But if database is critical and point in time recovery can be a requirement then database must be in FULL Recovery mode. Which is an indication for SQL Server that keep all the log in transaction log file and don't reuse, until it is backed up (transaction log backup).


  I've archived data from the database and then done a full backup...I
   was expecting the log to then be basically "empty". e.g.* Full backup or even Transaction Log backup don't left log file empty.
  Transaction log backup only mark log file space to reusable but still
  file size remains same.


If archiving means you have removed some data after fetching some data and keeping it in any other format. Then one must keep in mind that deleting data never reduce physical size of data or log files, though space could be available for reuse by database.