Carbon-14 dating assumes that the carbon dioxide on Earth today has the same radioactive content as it did centuries ago. If this is true, the amount of absorbed by a tree that grew several centuries ago should be the same as the amount of absorbed by a tree growing today. A piece of ancient charcoal contains only 15% as much of the radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal? (The half-life of Carbon-14 is 5715 years.)

It is impossible to predict when a particular atom will decay. However, it is equally likely to decay at any instant in time. Therefore, given a sample of a particular radioisotope, the number of decay events −dN expected to occur in a small interval of time dt is proportional to the number of atoms present N, i.e.
-(dN)/(dt)propto N
For different atoms different decay constants apply.
-(dN)/(dt)=\lambda N
The above differential equation is easily solved by separation of variables.
N=N_0e^(-lambda t)  
where N_0 is the number of undecayed atoms at time t=0.
We can now calculate decay constant lambda for carbon-14 using the given half-life.
N_0/2=N_0e^(-lambda 5715)
e^(-5715lambda)=1/2
-5715lambda=ln(1/2)
lambda=-(ln(1/2))/5715
lambda=1.21 times 10^-4
Note that the above constant is usually measured in seconds rather than years.
Now we can return to the problem at hand. Since the charcoal contains only 15% (0.15N_0 ) of the original carbon-14, we have
0.15N_0=N_0e^(-1.21times10^-4t)
Now we solve for t.
e^(-1.21times10^-4t=0.15)
1.21times10^-4=-ln 0.15
t=-(ln0.15)/(1.21times10^-4)
t=15678.68
According to our calculation the tree was burned approximately 15679 years ago.                                                                                                    
https://en.wikipedia.org/wiki/Radioactive_decay

https://en.wikipedia.org/wiki/Separation_of_variables