I have just found this Plugin that allows the user to write the Selenium IDE Log to a file. Please read this article for more information.
This plugin for Selenium-IDE saves log messages to a file in real time at a user selectable log level. Once this plugin is installed, a FileLogging tab will be added to the options dialog and a FileLogging menu will be added to the log pane.
This version brings a bunch of user experience improvements and some internal improvements. See the screen shot for details. The current settings are also saved and will be retained if you restart the Selenium IDE or Firefox.
Mini user guide for the plugin
Set the log file name and the log level from the Options dialog (FileLogging tab) of the Selenium IDE.
You can open the FileLogging tab of the Options dialog directly by selecting the Show options menu item from the FileLogging menu on the Log pane in Selenium IDE.
You can also set the log level for the log file directly from the FileLogging menu on the Log pane.
Set the log file name empty to stop writing to the log file.
The FileLogging menu on the Log pane will also show if FileLogging is on.
Download the plugin here