An ancient skull has a carbon- 14 decay rate of 0.85 disintegration per minute per gram of carbon \((0.85\) dis \(/ \min \cdot \mathrm{g} \mathrm{C})\). How old is the skull? (Assume that living organisms have a carbon-14 decay rate of 15.3 dis \(/ \mathrm{min} \cdot \mathrm{g} \mathrm{C}\) and that carbon- 14 has a half-life of 5715 yr. \()\)

Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting radiation. This happens spontaneously in nature, and it is the mechanism used in Carbon-14 dating to determine the age of artifacts.

In the context of Carbon-14, it's a type of carbon isotope that is radioactive, and it decays over time. All living organisms take in Carbon-14 from their environment, and when they die, they stop absorbing any new Carbon-14. From that point on, the Carbon-14 that is contained within the organism's remains will begin to decay at a predictable rate.

The rate at which radioactive substances decay is measured in half-lives—the time it takes for half of the material to decay. The half-life is a crucial concept because it provides a clock that can measure time from several hundred years up to several billion years. Because Carbon-14 has a well-known half-life, by measuring how much Carbon-14 is left in a sample compared to a living sample, we can estimate how long it has been since the organism's death.

The half-life calculation is essential when working with radioactive decay because it allows us to determine how long it takes for half of the original isotope to decay. For Carbon-14, this value is approximately 5715 years.

To calculate how long it's been since an organism containing carbon died, scientists compare the amount of Carbon-14 in the fossil to the amount of Carbon-14 in a modern day sample. A simple mathematical formula called the decay formula can express this relationship: \( N = N_0 \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} \), where \(N\) represents the final amount, \(N_0\) is the initial amount, \(t\) is the elapsed time, and \(t_{1/2}\) is the half-life of Carbon-14.

When solving problems involving half-life, one important tip is to make sure the units match. For example, if the half-life is given in years, the time of decay should also be expressed in years. This helps avoid errors and ensures accuracy in calculations.

Logarithmic functions play a critical role in interpreting the data obtained from radioactive decay and Carbon-14 dating. When the decay formula is rearranged to solve for time \(t\), logarithms are used to deal with the exponent in the equation.

To move forward from the formula \(0.85 = 15.3 \left(\frac{1}{2}\right)^{\frac{t}{5715}}\), taking the natural logarithm of both sides helps us extract the variable \(t\) from the exponent. The calculation looks as follows: \(\ln(0.85) = \ln\left( 15.3 \times \left(\frac{1}{2}\right)^{\frac{t}{5715}} \right)\), then using the properties of logarithms to simplify: \(\ln(0.85) - \ln(15.3) = \ln\left(\frac{1}{2}\right)^{\frac{t}{5715}}\), and after applying the logarithmic property that allows us to bring the exponent down, we get \(\frac{t}{5715} = \frac{\ln(0.85/15.3)}{\ln(1/2)}\), which can be solved for \(t\).

Understanding logarithmic functions is vital because they give us a tool to work backward from the observed decay back to the time of death—answering questions like 'How old is this ancient skull?' by converting the diminishing amounts of Carbon-14 into concrete units of time.