Cobalt-60 is used extensively in medicine as a source of \(\gamma\) -rays. Its half-life is \(5.27\) years. (a) How long will it take a \(\mathrm{Co}-60\) source to decrease to \(18 \%\) of its original activity? (b) What percent of its activity remains after 29 months?

The concept of half-life is critical to understanding radioactive decay. Imagine you have a collection of unstable atoms that spontaneously transform into different atoms over time. The half-life is the time required for half of these unstable atoms, known as radioactive isotopes, to decay. For example, if the half-life of a particular isotope is 5 years, after 5 years only 50% of the original amount will remain; the rest will have decayed.

When you're tackling exercises involving half-life, remember that every time one half-life passes, the remaining quantity of the substance will be halved. This pattern continues such that after two half-lives have elapsed, only 25% (half of 50%) remains, and so forth. The half-life is a constant time period and it doesn’t change regardless of how much material you have.

Exponential decay portrays how a quantity decreases rapidly at first, then slowly over time. It's governed by the equation \( A(t) = A_0e^{-\frac{t}{\tau}} \), where \( A(t) \) represents the amount remaining after time \( t \), \( A_0 \) is the initial amount, and \( \tau \) is the mean lifetime, which relates to the half-life by \( \tau = \frac{T}{\textrm{ln}(2)} \).

The concept of exponential decay is not exclusive to radioactive decay; it is also encountered in fields such as pharmacology, to describe how drugs metabolize in the body, and in finance, to depict depreciation of assets.

Radioactive isotopes, or radioisotopes, are variants of chemical elements that have an unstable nucleus and emit radiation as it moves toward a stable form. Each element can have several isotopes, differing in the number of neutrons in the nucleus. Only certain isotopes of an element are radioactive. For instance, Carbon-14 is a well-known radioactive isotope used in radiocarbon dating, while Carbon-12 is stable.

These isotopes have various applications, from dating archaeological finds to treating cancer and sterilizing medical equipment. Understanding the particular isotope you are dealing with, such as Cobalt-60 in our exercise, and its specific properties like half-life, is essential in mastering radioactive decay calculations.

Logarithms are powerful tools in solving radioactive decay problems. They enable us to find the time taken for a substance to decay by a certain amount or the proportion of the substance remaining after a period of time. The equation for decay based on half-life can also be expressed as \( A(t) = A_0 * \frac{1}{2}^{\frac{t}{T}} \), where \( T \) is the half-life. But solving for time \( t \) when you have a decay percentage requires logarithms.

The use of logarithms to isolate \( t \) in our original exercise exemplifies this: \( t = T * \frac{\textrm{log}(final\frac{amount}{initial amount})}{\textrm{log}(\frac{1}{2})} \). Logarithms essentially 'unpack' the exponential function, making it possible to find the exponent - which in the case of decay calculations represents the time elapsed or required.