A sample of wooden air craft is found to undergo 9 dpm \(g^{-1}\) of \(C^{14}\). What is approximate age of air craft? The half life of \(_{6} \mathrm{C}^{14}\) is 5730 year and rate of disintegration of wood recently cut down is 15 dpm \(\mathrm{g}^{-1}\) of \(_{6} \mathrm{C}^{14} ?\)

Radioactive decay is a fundamental process by which an unstable atomic nucleus loses energy by emitting radiation. In the context of carbon-14 ($C14$), it refers to the transformation of this radiocarbon isotope into a more stable nitrogen-14 ($N14$) nucleus over time. This process occurs at a predictable rate, characterized by the nuclide's half-life. When an organism dies, it stops absorbing $C14$, and the amount of $C14$ in its tissues begins to decrease through radioactive decay. By measuring the remaining $C14$, scientists can estimate the time since the organism's death, which is the principle behind carbon-14 dating.

Understanding radioactive decay enables us to estimate the age of ancient objects, such as the wooden aircraft in our problem. This natural clock provides invaluable time stamps for archaeological and geological materials, making it one of the most widely used tools in science for dating artifacts.

The half-life of a radioactive isotope is the time required for half of the radioactive atoms in a sample to decay. For carbon-14 ($C14$), this value is approximately 5730 years. A crucial aspect of half-life calculation is understanding that it is a constant value that does not depend on the size of the sample or its initial amount. Instead, it solely depends on the properties of the isotope itself.

In the calculation for the wooden aircraft's age, we use the half-life to determine how many years have passed for the disintegration rate to drop from 15 dpm/g to 9 dpm/g. Since half-life remains consistent, we can input it into calculations as a known variable to help solve for the age of our sample. This principle enables us to decode historical timelines, geologically date layers of earth, and even measure the decay in living systems.

Logarithmic equations are indispensable in chemistry, especially for analyzing processes that follow an exponential trend such as radioactive decay. The logarithm essentially allows us to solve for the power to which a number must be raised to get another number, which in the case of decay, helps to determine the time involved in the process.

Using logarithms, we can rearrange the decay formula to solve for the age of the wooden aircraft. By taking the natural log (ln) or log to base 10 of both sides of the decay equation, we isolate the variable representing time. This mathematical operation converts an exponential equation into a linear one that we can solve using algebra. In our exercise, it was necessary to employ a logarithmic equation to figure out how long the $C14$ had been decaying in the aircraft wood. This utilization of logarithms is a textbook example of applying mathematical concepts to solve practical problems in chemistry and beyond.