With my S530 I had to install the Logitech Control Center for OS X in order to configure the custom keys. Once you install that, you'll see a control center entry in your System Preferences panel:
Before you can assign a shortcut to the Home key on the keyboard, you need to make a keyboard shortcut for Launchpad as there is none by default. Open System Preferences and go to Keyboard settings. Click the Keyboard Shortcuts tab and select Launchpad & Dock. Make sure 'Show Launchpad' is checked and then double click over the area that shows the shortcut to assign a new shortcut (I used Alt-Cmd-L in the screen shot). If you can't assign a shortcut try unchecking and re-checking the Show Launchpad setting:
Now enter the Logitech Control Center, select your device and click the Configure button:
Click the Keys icon at the top to configure custom keys on the keyboard. Find the Home key in the list. Set the 'Assigned action' for the Home key to 'Keystroke. And configure the keystroke issued to be the one you assigned to Launchpad above. In my case it would be Alt-Cmd-L.