Exponential Decay - Example
Problem: A radioactive isotope of iodine decays exponentially. There is 50 mg of the element initially, 35 mg after 4 days.
(a) Find a formula of the form
(b) Find the half-life, , of the isotope.
As we found out, the formula decsribing the decay is
Half-life is the time, , after which half of the initial amount of 50 mg decomposes, and 25 mg is left. Hence, to find we have to solve for the equation
We divide both sides by 50, and take the natural logarithm of both sides. We obtain
Since (from properties of logs), we get
Hence, the half-life is
The latter gives the half-life days, approximately. As we know, that means that half of the amount will be gone after 7.78 days; after the next 7.78 days half of what's left will be gone again, and so on.