Give this a spin:
BEGIN TRY
BACKUP LOG @v_DatabaseName TO DISK = @v_BackupFileLocation WITH NAME = @v_BackupFileName, SKIP, STATS = 10
END TRY
BEGIN CATCH
Set @v_ErrorMailSubject = 'Transaction log backup failure in database '+@v_DatabaseName
Set @v_ErrorMailText = 'An error occured in transaction log backup for database '+@v_DatabaseName+'.'+Char(13)+Char(13)
Set @v_ErrorMailText = @v_ErrorMailText + 'Error '+Cast(Error_Number() as nvarchar)+' - ['+Error_Message()+']'
exec msdb.dbo.sp_send_dbmail @recipients=@v_ErrorMailRecipient,
@subject = @v_ErrorMailSubject,
@body = @v_ErrorMailText
END CATCH
You'll need to configure the database email configurations first, though