How have geologists determined the age of earth?
By examining the isotopic abundances of meteor fragments.
The search for really old rocks frustrated scientists for a long time. The problem is that the earth's surface is constantly eroding as well as turning over due to tectonic plate movement. Surface material sinks into the core and new lava flows up to cover the surface.
In the 1950's a clever scientist thought that the age of the earth could be estimated by looking at the raw material from which the earth formed. That material is still falling to the earth as meteorites.
An age of 4.56 billion years was calculated from measurements of the relative abundance of various isotopes in the Canyon Diablo iron meteorite. This remains the accepted age of the earth.
Here's a little more detail on that story: How is Earth's Age Calculated?
Also look for the book: The Age of the Earth, by G. Brent Dalrymple