The exponential formula for the half-life of a radioactive isotope is y=y0ekt, where y is the amount of the isotope remaining after t years, y0 is the initial amount of the isotope, k is the decay constant, and e is the transcendental number approximately equal to 2.71828.

See explanation below.

Explanation:

For this case we atart fom the proportional model given by the following differential equation:

And if we rewrite this expression we got:

If we integrate both sides we got:

And using exponential on both sides we got:

Where represent the initial amount for the isotope and t the time in years and A the amount remaining.

If we want to apply a model for the half life we know that after some time definfd the amount remaining is the hal, so if we apply this we got:

We can cancel and we got:

If we solve for k we can apply natural log on both sides and we got:

And that would be our proportional constant on this case.

If we replace this value for k int our model we will see that:

And using properties of logs we can rewrite this like that:

And thats the common formula used for the helf life time.