edit: To answer the question directly, yes "logging" is a valid word to use. You generally "log in" but while you are in the process of that act it would be called "logging in".
In the case of how you know logging to be used, it is the verb form of the noun log as in a segment of a tree. Computing borrowed the term "log" in reference to logbooks that are used to track events. Originally when you would "log in" or "log on" (which historically was two different things but nowadays "log on" is not used as much) this action was recorded much like in logbooks since administrators needed to manage time that users were on the system. Logging also refers to any output that is generated internally within the system rather than being sent back to the user and is often stored in a "log file" which is often shortened to just "log" or the plural "logs."
Just in case anyone's interested in "in" versus "on" it referred to how much access was given to a system. Logging "on" meant that you had specific actions you could make on the system where as logging "in" meant that you had root (admin) access and could perform any function. Somewhere along the line "in" pretty much became the standard and the distinction blurred. When terminating a session where you logged in you would log out and where you logged on you would log off.