Did you try Logitech's SetPoint software? It may not let you change what program the key opens, but it's worth a try. I don't have a Logitech keyboard, so I can't check.
You could also try replacing the Calculator executable (%windir%\system32\calc.exe) with the other calculator, but that may be risky.
Finally figured out a solution!
UPDATE 2014
This has been tested on Windows 8 and 8.1 and works.
Using a VBS file will make the calculator instance keep focus.
Dim oShell
Set oShell = WScript.CreateObject ("WScript.Shell")
oShell.run "C:\Windows\System32\calc.exe", 0, False
Set oShell = Nothing
Just create a text file like calculator.vbs and copy/paste the above code and double click or link the file to your keyboard button.
[ORIGINAL ANSWER]
Hopefully this isn't specific to a version of IntelliType. However I'm using IntelliType 8.2 x64 for Windows 7 with my Natural Ergonomic Keyboard 4000.
Once you have installed IntelliType go to Control Panel -> Keyboard then Key Settings tab.
Find the Calculator button in the Key Name column and click Configure. It will be defaulted to "Starts Calculator" now choose "Start a program, Web page, or file" and locate your copy of calc.exe which should be in your system32 folder. In my case it's "C:\Windows\System32\calc.exe"
Click "Finish" and it's now assigned and working as it did on previous Windows.
Now when you click the calculator button it will open multiple calculators instead of just bringing focus to an existing one.
As part of owning a business I do most of the bookkeeping. And having the ability to rapidly open multiple calculators with the calc button is great!
Best Answer
There is a keyboard shortcut. In Scientific mode, N gives you natural log. Lots of other shortcuts here: Microsoft Support.
Another option is to download the free Calculator Plus from the Microsoft store. That has dedicated buttons for an extensive range of things, including ln:
BTW, it also has an option for the "Classic View", which looks like the earlier Windows calculators.