How are initial radioactive isotope quantities assumed?

[Ty1erC]

I'm stuck on this idea. How are initial radioactive isotope quantities assumed in radiometric dating? There are current abundances for all isotopes, but wouldn't these abundances have been different in the past (much higher)? I honestly can't grasp how radioactive isotopes with short half lives can even exist today, given the age of the Earth. Can someone set me straight-- Google isn't cutting it.

Thank you.

 

[mfb]

It depends on the isotope used.

Sometimes the initial concentration of a decay product is zero (because one type of atom gets included in a crystal but a different type does not).
Sometimes the initial fraction of a radioactive isotope is (nearly) constant over time because it is in equilibrium between being produced by cosmic rays or uranium decays and its own decay.
Sometimes you can reconstruct how many decays happened since formation by radiation damage in materials.

The Wikipedia articles for the various dating techniques should describe what is used to estimate the initial concentrations.