Elk have been taken with old cartridge's like the 25-20 and 22 Savage High Power, but those were in the days of my grandad and every shot had to bring something home for dinner, it is all about WHERE the bullet strikes the animal!
I live in the center of every big game animals territory in Idaho. We are out and about with cows etc and find carcasses of animals lost by hunters. When skinning my wife's trophy whitetail buck last year we found a 25ca bullet between the skin and rib, NO penetration! The tip was barley dented. So besides shot placement do not try to "stretch the barrel". It seems to be too common.
Do not shoot light bullets thinking speed always means more energy. A heavier bullet works best on elk sized game.
When I guided years ago the minimum for elk was 30ca rifle with 180gr or 154gr in a 7mag.

This is simple enough for newbie elk hunters.