It is certainly possible and should indeed be done.
However, your question is very broad indeed as there are many uses for data centres ranging from ones that cater for a single customer through to global cloud data centres. You need to be more specific.
One of the major issues with logging in DCs is obviously the amount of traffic generated and this can be a real headache. Busy DC's will generate MB of data per hour or more which has, somehow to be handled.
So while it is possible, it isn't necessarily easy and some DC's will limit the traffic in certain logs.
There is also the issue of what to do with the data. Having it is one thing, using it is totally another. You absolutely have to have systems that summarise and chart the data so that humans can spot issues and react to them in a reasonable time.
You may note that, depending on what the DC is used for and also what country it is in, there may also be legal requirements for logging.