First, it is important to note that the half life of carbon-14 is 5730 years. This says that the amount of carbon is exponentially related to the length since decay began. To start, note that half life is represented by equation 1 below:
This says that the amount left is equal to the initial amount multiplied by e raise to the power of the k (the decay constant) and the number of half lives, or simply t/t1/2; where t is the time (age) and t1/2 is the half lift of the subject.
If you are unfamiliar with the k constant for carbon-14, then it can be solved by plugging in a known value: the time of the half life, or t/t1/2=1. Since the time you plug in at this point is the half life of the substance, then A(t)=(1/2)*A.
Then you have:
solving you get:
To find k, you simply take the natural log (ln) of both sides.
Now that you have k, it is a matter of substituting the amount of carbon-14 left at the age you would like to calculate.
Notice that the A's cancel out and you get:
.46=e-.693t/5730 or .46=e-.00012t
Take the natural log (ln) of both sides to eliminate e and you have
Solving for t you get:
t=-ln(.46)/.00012 = 6471 years