Oracle writes the alert.log file to the directory as specified by the BACKGROUND_DUMP_DEST parameter. If this parameter is not set, the alert.log will be created in the ORACLE_HOME/rdbms/trace directory.
SQL> show parameter BACKGROUND_DUMP_DEST
NAME TYPE VALUE
------------------------------------ ----------- ------------------------------
background_dump_dest string /app/oracle/diag/rdbms/o11gr1/o11gr1/trace
With the right permission, you can try going to Database menu within Toad | Administer | Oracle Parameters and double click on the background_dump_dest parameter and check it via the wizard or run the alter system command to change the directory. Please see Oracle documentation for alter system command.
