Starting in MySQL 5.7, the mysql command accepts the --syslog option. So I upgraded a test system to MySQL 5.7, and it works.
I will make an alias from mysql to mysql --syslog
From MySQL 5.7 Reference Manual - mysql Logging
syslog Logging Characteristics
If the --syslog option is given, mysql writes interactive statements
to the system logging facility. Message logging has the following
characteristics.
Logging occurs at the “information” level. This corresponds to the
LOG_INFO priority for syslog on Unix/Linux syslog capability and to
EVENTLOG_INFORMATION_TYPE for the Windows Event Log. Consult your
system documentation for configuration of your logging capability.
Message size is limited to 1024 bytes.
Messages consist of the identifier MysqlClient followed by these
values:
SYSTEM_USER
The system user name (login name) or -- if the user is unknown.
MYSQL_USER
The MySQL user name (specified with the --user option) or -- if the
user is unknown.
CONNECTION_ID:
The client connection identifier. This is the same as the
CONNECTION_ID() function value within the session.
DB_SERVER
The server host or -- if the host is unknown.
DB
The default database or -- if no database has been selected.
QUERY
The text of the logged statement.
Here is a sample of output generated on Linux by using --syslog. This
output is formatted for readability; each logged message actually
takes a single line.
Mar 7 12:39:25 myhost MysqlClient[20824]: SYSTEM_USER:'oscar',
MYSQL_USER:'my_oscar', CONNECTION_ID:23, DB_SERVER:'127.0.0.1',
DB:'--', QUERY:'USE test;' Mar 7 12:39:28 myhost MysqlClient[20824]:
SYSTEM_USER:'oscar', MYSQL_USER:'my_oscar', CONNECTION_ID:23,
DB_SERVER:'127.0.0.1', DB:'test', QUERY:'SHOW TABLES;'