There's no way to export logs to PDF, you can save them in the MHT, HTML and XML formats (see the "Exporting Test Results" help topic). Also, you can print logs (see the "Printing Test Results" help topic).
If you want to write logs to your custom format during the test, you need to handle log events (OnLogMessage, OnLogError, etc.) and save data to an external file from their handlers. See the "About Events and Event Handling" help topic.