Age samples using radioactive isotopes.
Isotopic dating of rocks works from the parent-to-daughter ratio rather than from an activity measurement, because over geological timescales the daughter accumulates in the mineral lattice. The original parent population is the parent still present plus the daughter it produced, so the surviving parent fraction follows directly and the number of elapsed half-lives is its base-two logarithm. Multiplying by the half-life gives the age.
Parent-daughter isotopic age
original parent = parent remaining + daughter formed; half-lives = log2(original / remaining); age = half-lives x half-life
Potassium-argon and argon-argon at 1250 Ma suit volcanic rocks from tens of thousands to billions of years. Uranium-lead at 4470 Ma is the workhorse for the oldest rocks. Rubidium-strontium and samarium-neodymium cover long-lived crustal and mantle histories.
Three: no daughter isotope was present when the mineral formed, no parent or daughter has been lost since, and the half-life is known accurately. Real work tests the first two with isochron plots across several minerals from the same rock.
Its 5730-year half-life exhausts the measurable signal after about 50,000 years, and rocks contain no atmospheric carbon to start the clock. Long-lived isotopes locked into crystal lattices are needed for geological ages.