**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

that models the process of decay.

(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.