An amount of M radioactive isotope will have Mn = M0
´ (0.5)n left, in which n =
age/t1/2 and is the number of half-lives.
Thus, (Mn/ M0) = 0.5n,
which can be converted to the following.
log(Mn/ M0) = log0.5n = n ´ log0.5 = (age/t1/2) ´ log0.5, or
log(Mn/ M0) = (age/t1/2) ´ log0.5
Therefore, if you know the initial and the final amount (or the ratio Mn/ M0) and know the half-life, you can find the time (age) from the beginning of the decay to the moment the object is tested.
For example, if an amount of 8% C-14 is left in a piece of animal fossil, you just need to solve the following equation to find when did the animal die.
Mn/ M0 = 8% = 0.08, and t1/2 of C-14 is 5730 years,
since log(Mn/ M0) = (age/t1/2) ´ log0.5
log0.08 = (age/5730) ´ log0.5
age = (log0.08/log0.5) ´ 5730 = 20880 years