In some cases, you will need to change the product key for Microsoft Office 2010 or 2013. Unfortunately, there is no way to change it from Office programs via GUI, but there are still different ways to do it.
Personally, I prefer to do it from command prompt. I think that this way is transparent and clear to do all steps clearly and under control. To do this, you must open command prompt window as administrator and navigate to the one of this folders (depends on witch version of Office and Windows you have):
C:\Program Files (x86)\Microsoft Office\Office14
C:\Program Files\Microsoft Office\Office14
C:\Program Files (x86)\Microsoft Office\Office15
C:\Program Files\Microsoft Office\Office15
In this folder, there is situated the VBS script OSPP.VBS. This script is dedicated to work with product key and to change the product key you have to use syntax CSCRIPT OSPP.VBS /INPKEY:KEY_STRING (where KEY_STRING is the key). After you have changed the key, you have to activate it and this is possible to do it with the same script: CSCRIPT OSPP.VBS /ACT
Of course the script has more than this functionality and you can view them by typing CSCRIPT OSPP.VBS /?.