All object that have a blue square around them on the
star map have a log file. A log file is a message from something that you own, like a ship, base or pod. A log message is written to you any time something of importance happens, like a riot or a ship has its cargo stolen. When an object that has a log event is selected as the current hammer or anvil object you will see the log button on the space command panel appear.

Pressing one of the green buttons on the right side of the display will list all the logs for objects of the selected type. The wings button is a different color because there are currently no log files for fighter wings.