The error log contains information indicating when mysqld was started and stopped and also any critical errors that occur while the server is running. If mysqld notices a table that needs to be automatically checked or repaired, it writes a message to the error log.
On some operating systems, the error log contains a stack trace if mysqld dies. The trace can be used to determine where mysqld died. See MySQL Internals: Porting.
You can specify where mysqld writes the error
log with the
--log-error[=
option. If no file_name]file_name value is
given, mysqld uses the name
by
default and writes the file in the data directory. If you
execute host_name.errFLUSH LOGS, the error log is renamed
with the suffix -old and
mysqld creates a new empty log file. (No
renaming occurs if the --log-error option was
not given to mysqld.)
If you do not specify --log-error, or (on
Windows) if you use the --console option,
errors are written to stderr, the standard
error output. Usually this is your terminal.
On Windows, error output is always written to the
.err file if --console is
not given.
The --log-warnings option or
log_warnings system variable can be used to
control warning logging to the error log. The default value is
enabled (1). Warning logging can be disabled using a value of 0.
If the value is greater than 1, aborted connections are written
to the error log. See Section B.1.2.11, “Communication Errors and Aborted Connections”.
If you use mysqld_safe to start
mysqld, mysqld_safe
arranges for mysqld to write error messages
to a log file. If you specify a filename via
--log-error to mysqld_safe
or mysqld, that filename is used. Otherwise,
mysqld_safe uses the default error log file.
If mysqld_safe is used to start
mysqld and mysqld dies
unexpectedly, mysqld_safe notices that it
needs to restart mysqld and writes a
restarted mysqld message to the error log.

User Comments
I did some testing with MySQL 4.0.21 this morning. Here's a typical snippet from my "hostname.err" file. To generate this, I did a "NET START MySQL", connected with one session and ran a 2000-entry query, and then did a "NET STOP MySQL" while the query was still returning data.
MySql: ready for connections.
Version: '4.0.21-nt-log' socket: '' port: 3306 Source distribution
040923 10:00:00 MySql: Normal shutdown
040923 10:00:01 MySql: Forcing close of thread 1 user: 'root'
040923 10:00:01 InnoDB: Starting shutdown...
040923 10:00:03 InnoDB: Shutdown completed
040923 10:00:03 MySql: Shutdown Complete
The Windows Application Event Log recorded 3 messages at the same time. All of the messages corresponded with the entries prefixed with the "MySQL:" entries in the hostname.err file. (OK)
However, all 3 messages were logged as ERRORS; this designation is misleading. If anything, the "Normal Shutdown" and "Shutdown Complete" messages should have been logged as INFORMATION and the "Forcing close of thread..." message should have been logged as a WARNING.
Also, it is important to note that the MySQL service startup was NOT LOGGED in the Event Log.
Long story short, if you are a Windows user, it is probably still best (as of 4.0.21) to stick with your existing "parse-the-.err" script rather than rely on the Windows Event Log if you're interested in MySQL service starts, stops and abnormal events.
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