Carbon dating logarithmic
As soon as a living organism dies, it stops taking in new carbon.The ratio of carbon-12 to carbon-14 at the moment of death is the same as every other living thing, but the carbon-14 decays and is not replaced.The carbon-14 decays with its half-life of 5,700 years, while the amount of carbon-12 remains constant in the sample.
I will give you the setup of the problem and leave the algebra to you. We know at $t = 0$, we have 0\%$ of the Carbon, right? So then we know at $t = 5750$, there is \%$ of the carbon remaining. That gives you enough information to solve for $r$. Plug in your result for $r$ from the previous equation and solve for $t$.
We end up with a solution known as the "Law of Radioactive Decay", which mathematically is merely the same solution that we saw in the case of light attenuation.
We get an expression for the number of atoms remaining, N, as a proportion of the number of atoms N, where the quantity l, known as the "radioactive decay constant", depends on the particular radioactive substance.
Again, we find a "chance" process being described by an exponential decay law.
We can easily find an expression for the chance that a radioactive atom will "survive" (be an original element atom) to at least a time t.