Sql-server – What ‘resets’ the transaction log

I'm trying to create a backup plan which creates a full backup twice a day.
A differential backup every hour and a log backup every 5 minutes.

It works fine but I've noticed the log backup (.trn file) grows and grows in size each time.

If I use backup log with init it overwrites and I'm getting the too recent error.

I thought that after another full or differential backup the next log backup would only include the log back-ups "since the last full / diff backup"?

BACKUP DATABASE 
    lagerdb 
TO DISK = N'C:\SQLBACKUP\lagerdb-rechts.bak' 
WITH INIT, NOUNLOAD, NAME = N'lagerdb full', NOSKIP, STATS = 20, NOFORMAT

BACKUP DATABASE 
    lagerdb TO DISK = N'c:\SQLBACKUP\lagerdbd-rechts.bak' 
WITH INIT, NOUNLOAD, NAME = N'lagerdb differential', NOSKIP, STATS = 20, NOFORMAT, DIFFERENTIAL 

BACKUP LOG 
    lagerdb 
TO DISK = N'C:\SQLBACKUP\lagerdb-rechts.trn' 
WITH NAME = N'lagerdb logs'

I did something with the suggestion below to backup the log to separate files with a datetime stamp:

 -- Declares a string
 declare @sql nvarchar(255)
 declare @ParmDefinition NVARCHAR(500)

 -- declares the path name. Seperated by disk drive and directory.
 declare @bkName varchar(255)

 -- Declares the filename. 
 declare @bkdbName varchar(50)
 declare @bkdbDateTime varchar(20)

 --Get Timestring
 set @sql = N'select @parm1OUT = CONVERT(varchar(20),CONVERT(DATE,GETDATE(),104),104)     + ''_'' + Substring(CONVERT(varchar(8),getdate(),114),1,2) + Substring(CONVERT(varchar(8),getdate(),114),4,2)'
 SET @ParmDefinition=N'@parm1OUT varchar(20) OUTPUT'

 exec sp_executesql 
 @sql, 
 @ParmDefinition,
 @parm1OUT=@bkdbDateTime OUTPUT

 set @bkdbName = 'LagerDB_' + @bkdbDateTime

 -- Enter the Drive letter below. Please remove C: as current value.
 set @bkName = 'C:'

 -- Enter the folder name below (with lead and end '\' back slashes.
 -- Ensure that you change the folder from Temp below.
 set @bkName = @bkName + '\SQLBackup\'
 set @bkName = @bkName + @bkdbName 

 -- Sets the backup running by executing the @sql string.
 set @sql = 'BACKUP DATABASE lagerdb TO DISK = '''+@bkName+'.bak'''

 print @sql
 exec sp_executesql @sql

(e.g.: LagerDB_06.12.2011_1418.bak)

But the answer I was really looking for is that I thought it would reset after a full backup. Maybe I should schedule a backup with init after a full or differential backup?