Suppose you find a piece of ancient pottery and find that the glaze contains radium, a radioactive element that decays to radon and has a half-life of 1,620 years. There could not have been any radon in the glaze when the pottery was being fired, but now it contains three atoms of radon for each atom of radium. How old is the pottery?

Radioactive decay is a random process where unstable atomic nuclei lose energy. The **half-life** of a radioactive substance is the time it takes for half of the radioactive atoms in a sample to decay. In our pottery example, radium has a half-life of 1,620 years. This means it will take 1,620 years for half of the radium atoms to decay into radon.

The half-life formula is:
\[ N(t) = N_0 \times \frac{1}{2}^{t / T} \]
Where:

• \(N(t)\) is the number of radium atoms remaining at time \(t\)
• \(N_0\) is the initial number of radium atoms
• \(T\) is the half-life (1,620 years in this case)

This formula shows how the remaining number of atoms decreases exponentially over time.


You should note that during each half-life, only a fixed fraction (half) of the substance decays, regardless of the initial amount. This is what defines the exponential nature of radioactive decay.

The term **exponential decay** describes a process in which the amount of a substance decreases rapidly at first, but then more slowly over time. It’s characterized by the property that in each equal time interval, the substance reduces by the same fraction. The equation for exponential decay we used in this problem is:
\[ N(t) = N_0 \times \frac{1}{2}^{t / T} \]
In this equation, the base of the exponent (\( \frac{1}{2} \)) indicates that we are halving the remaining amount after each period \(T\) (the half-life).


**Visualizing Exponential Decay:**
Imagine you have 100 atoms to start with. After one half-life (1,620 years for radium), you'd have 50 atoms left. After another 1,620 years, you'd have 25 atoms left, then 12.5, and so on.


This kind of decay is contrasted with linear decay, where the amount decreases by a fixed quantity each interval. With exponential decay, the speed of decay slows down as fewer and fewer atoms remain.
**Practical Application:** Radioactive dating, like in this exercise, makes use of this concept to determine the age of archaeological finds by measuring the ratios of remaining radioactive material and its decay products.

The **natural logarithm** (ln) is a logarithm to the base \(e\), where \(e\) is approximately 2.71828. It’s often used in calculations involving exponential growth or decay due to its unique properties that simplify these types of equations.

In this problem, we used the natural logarithm to solve for time \(t\):

1. We started with the decay equation:
   \[ \frac{1}{4} = \frac{1}{2}^{t / 1620} \]
2. Took the natural logarithm of both sides:
   \[ \text{ln} \frac{1}{4} = \text{ln} \frac{1}{2}^{t / 1620} \]
3. Applied the logarithm power rule, which states \( \text{ln}(a^b) = b \text{ln}(a) \):
   \[ \text{ln} \frac{1}{4} = \frac{t}{1620} \text{ln} \frac{1}{2} \]

By using the properties of logarithms, we simplified the equation, making it easier to solve for \(t\).

The natural logarithm helps convert exponential equations into linear ones, which are easier to manipulate algebraically. In general, natural logarithms are extremely useful when dealing with problems related to exponential processes, such as radioactive decay, population growth, and certain financial calculations.