I am curious to know how many miles you have driven in those 5 weeks, because 2 gallons of oil is a HUGE amount of leakage/consumption.
The one thing in which you can take some comfort is the fact that this engine uses a timing chain, rather than a belt. Based on your description, it sounds like there is a leak in the seal around the timing chain cover. However, 2 gallons in 5 weeks sounds like an Exxon Valdez-size leak, which seems contradictory in view of your two statements that you see "no leaks under the car", coupled with "there is oil leak around that area".
It is very possible that the oil leak is only apparent while the engine is running and the oil pump pressurizes the oil, so you might want to observe the area around the chain cover while the engine is idling. Pull on the throttle linkage a bit to speed-up the RPMs and see if the leak becomes more visible.
Bear in mind that the car's catalytic converter would likely prevent you from seeing the classic "blue oil smoke" from oil that is being burned in the engine. Or, more correctly the cat will do so until it is destroyed by having to process oil for a continued period of time. Thus, the sooner you can figure out and resolve the problem, the sooner you can possibly save the cat and reduce your repair expenses.
My theory at this point is that the engine in this car was very badly abused by the previous owner(s), and that you have both oil leaks and excessive oil consumption as a result of (pick one or more):
Lack of maintenance (excessively long oil change intervals and/or a clogged PCV system)
Allowing the engine to operate with a low oil level
Overheating that warped the cylinder head and scored the cylinders and/or led to a breached head gasket
I would suggest that you have a compression test and a leak-down test performed a.s.a.p. so that you can decide how to proceed with this car. If the engine is in as bad a condition as I suspect, it would be cheaper to get a "junk yard engine" than it would be to tear the present engine apart and fix all of the problems lying in the engine.
If this engine went for excessive periods without oil changes, even the timing chain may be living on borrowed time, so the sooner you get to the bottom of this problem the sooner you can decide how to proceed. Unfortunately, it appears that some unscrupulous person sold you badly-damaged goods and all you can do at this point is to limit the financial hit that you are going to take with repair or replacement of the car.
I sincerely wish you good luck, and I urge you to have your next car carefully inspected by a mechanic of your choice prior to purchase.