A cross country flight does not require a landing, and yes, many of the scenic flights are cross country flights between Las Vegas, North Las Vegas, Henderson, and Boulder City, to Grand Canyon, and Canyon West, as well as Marble Canyon, Page, Flagstaff, Cliff Dweller's Lodge, Whitmore Ranch, and several other locations in the area.

The basic element of a cross country is a landing at a point other than the point of departure, except for cross country for the purposes of an ATP, which only requires that the flight reach a point more than 50 miles from the point of departure (but need not land). Cross country as applicable to the student pilot certificate is 25 miles with a landing, and for the private and commercial 50 nm with a landing.

To answer your question about logging cross country...just the same as any other.

Who logs cross country any more? I don't think I've put cross country in a logbook in over 25 years.