Strictly speaking, under the fourth Amendment, there is no "exemption." I don't think I have to quote it verbatim, we all know what it says. If they didn't have PC to swear out a warrant, they had none to enter.
On the other hand, if the guy had been shouting at the top of his lungs, making audible threats, or if someone inside had been calling for help, that gives LE "reasonable suspicion" to enter. If upon knocking, the man opened the door, and they saw something suspect in plain view, that would also give them reasonable suspicion.
To quote Sir William Pitt: "The poorest man may, in his cottage, bid defiance to all the forces of the Crown. It may be frail, it's roof may shake; the wind may blow through it; the storm may enter; the rain may enter; but the King of England may not enter; all his force dares not cross the threshold of the ruined tenement." It is this statement (spoken before the British House of Commons), and others like it, that are the basis for the Fourth Amendment.
There is no excuse, they didn't do their due diligence. That's on them. Were it up to me, that chief would be out of a job.