How to write custom messages to the alert log file

exec sys.dbms_system.ksdwrt(2,'Your message here')

The first argument can be either

1 -> Write to the trace file

2 -> Write to the database alert log file

3 -> Write to both of the above files

How to use adrci to show all messages related to shutdown and startup of a database

These commands will give you all messages related to a shutdown and a startup of an oracle database, respectively:

adrci> show alert -p "MESSAGE_GROUP LIKE 'shutdown'"
adrci> show alert -p "MESSAGE_GROUP LIKE 'startup'"

Example output for a shutdown:

Instance shutdown complete (OS id: 4058731)
2022-03-10 18:00:18.042000 +01:00
License high water mark = 863
2022-03-10 18:00:19.115000 +01:00
Instance shutdown complete (OS id: 3611031)
2022-03-10 18:00:33.286000 +01:00
Shutting down instance: further logons disabled
2022-03-10 18:00:36.357000 +01:00
License high water mark = 1
Shutting down archive processes
Archiving is disabled
2022-03-10 18:00:37.519000 +01:00
Shutting down archive processes
Archiving is disabled
2022-03-10 18:00:43.753000 +01:00
Instance shutdown complete (OS id: 3611306)

For the startup sequence, adrci will show you all the details of the startup, in other words what you normally see if you put a tail on the alert log. This is a rather lengthy output, so I am shorting it down. It should be familiar to most DBAs:

2022-03-10 18:00:22.413000 +01:00
Oracle instance running with ODM in PGA: Oracle Direct NFS ODM Library Version 6.0
2022-03-10 18:00:25.600000 +01:00
LICENSE_MAX_SESSION = 0
LICENSE_SESSIONS_WARNING = 0
Using LOG_ARCHIVE_DEST_1 parameter default value as USE_DB_RECOVERY_FILE_DEST
Autotune of undo retention is turned on.
IMODE=BR
ILAT =249
LICENSE_MAX_USERS = 0
SYS auditing is enabled
Oracle Database 19c Enterprise Edition Release 19.0.0.0.0 - Production
Version 19.14.0.0.0.
ORACLE_HOME:    /sw/oracle/product/19c
System name:    Linux
Node name:      myserver.mydomain.com
Release:        4.18.0-348.12.2.el8_5.x86_64
Version:        #1 SMP Mon Jan 17 07:06:06 EST 2022
Machine:        x86_64
Using parameter settings in server-side spfile /sw/oracle/product/19c/dbs/spfiletestdb01.ora
System parameters with non-default values:
etc etc

Since the output is very long, if you simply want to check for an indication of a database startup you could use this command instead:

adrci> show alert -p "message_text like '%Starting ORACLE instance%'"

which will give you a shorter list to examine, and in my case showing that the database was restarted Jan 28th and March 10th:

2022-02-28 07:46:54.505000 +01:00
Starting ORACLE instance (restrict) (OS id: 4058595)
2022-02-28 07:47:39.567000 +01:00
Starting ORACLE instance (normal) (OS id: 3300)
2022-03-10 18:00:22.421000 +01:00
Starting ORACLE instance (restrict) (OS id: 3611160)
2022-03-10 18:02:06.831000 +01:00
Starting ORACLE instance (normal) (OS id: 3429)