We have a 5 ft. diameter log that we wish to cut it in half and put it on our horizontal bandsaw, which has an opening width of approximately 30 inches.

Can we lift it on one side, then cut the top off and keep cutting down, the way you would quartersaw wood, but here you would get a wider piece, rather than, normally quartering the whole log?

If you want to truly quarter saw it, then you have to quarter the log. Chainsaw and wedges will get it quartered. There are several references on Youtube.

Kenneth, the answer to your question depends on whether you want the figure to show up (which means the rings must be around 80 to 90 degrees to the face). Some times quartersawing refers to lumber with 45 to 90 degree angles. In all cases, there is a question on how much of a piece of lumber has to show figure or has the indicated angles. We almost always see quartersawn and rift lumber sold together, where rift has 45 to 90 degree but does not show figure.

The most figured lumber will occur with certain sawing patterns and even with a few wedge-shaped pieces that are scrap.

It is certainly a standard, common practice to to saw a large log into quarters first. Some small sawmill manufacturers have a quartersawing attachment.

What I do, is rip it down the middle first leaving 2 equal halves. But then rather than using the chainsaw to saw down the middle of those halves, rendering quarters, which messes up the best two boards in each half. I usually move off center about 6" - 10" for my next chainsaw cuts. This will be the side down on the band mill. Then the first cut on the mill will be about the same distance above the pith, then I set that cant aside. Then I cut that main cant as it sits on the mill. This will start and end up with some nice rift lumber toward the top and bottom, and the best QS in the middle. But you could move right to cutting near the pith getting the best QS in the middle, then tipping up the edge of those cants with wedges for more QS.

The cuts on the other cants will be obvious.

Oh, one last thought. Usually a log of that size will have root buttresses which may make it somewhat square on up the first log. the definition for Medullary rays says they radiate out from the pith. However on a larger out of round log, it might be better describes as perpendicular to the cambium. So they can bend outward toward the surface in a buttress. This can end up rendering some partial QS as your finishing up the two cants from the sides of the two halves. Making your first halving cut with the chainsaw through the thinnest part of the log usually takes advantage of what I'm trying to share, so don't try to overthink it if you don't understand that.