If you write out your logs to a file inside the App's Documents directory, and you have access to the device, then you can download the Application Data directory. I've done this before to copy off the logs and SQLite database from our application. You can follow the instructions here:
Managing Application Data
To make a copy of your application’s device–based file system to your Mac:
- In Xcode, choose Window > Organizer.
- In the Organizer, select your device in the Devices list.
- In the Summary pane, click the disclosure triangle next to your application.
- Click the download button (the down-pointing arrow to the left of the Application Data package), as shown in Figure 3-7.
- In the dialog that appears, choose a location for the file-system copy.
Note: I'm not sure it show the triangle you need to click on, so I've included my own screenshot of it.

Now, if you don't have access to the device. Then the easiest way it to have a webserver that can capture the data, and have the application upload the data via a preference pane button.