When I executed Update-SPSolution command in PowerShell  to update  one of WSP-solutions, I got the error, that 'LOG_BACKUP' is full "Update-SPSolution : Das Transaktionsprotokoll für die 'SP19_Config'-Datenbank ist aufgrund von 'LOG_BACKUP' voll". In English versions of SharePoint the text of the error can be like this "The transaction log for database ‘SP19_Config’ is full due to LOG_BACKUP".

In this case the quickest solution is to Shrink database logs for the database 'SP19_Config' (Config database of SharePoint farm). You can do it manually with SQL Management Studio, but I prefer SQL request. You need to have very high level of permissions, close to DB admin.

use SP19_Config
go
alter database SP19_Config set recovery simple
go
dbcc shrinkfile('SP19_Config_log',10000)
go
alter database SP19_Config set recovery FULL
go

If you don’t have specific permissions, ask your DBAs about it.