Can't do a full backup because the log is full
I assume you don't have space in your drives to take a log backup. You can use the following query to trick SQL server into thinking that you are actually saving the log backup to disk, But you're not.

BACKUP LOG dbname TO DISK = 'NUL:';


Also if you're running SQL server 2005 or less, You can use the below query

BACKUP LOG dbname WITH TRUNCATE_ONLY;


Once the query completes you can go ahead and shrink the log file as usual using 

USE dbname;
DBCC SHRINKFILE(fileid , 0);


Also do note that the above backup query will break your log backup chain and your log-shipping setup, if you have one.