We can see that this formula is generally useful for artifacts that are up to 40,000 years old. For older artifacts, the amount of carbon-14 left is so small that small errors in sampling can create such large errors in age that this method is considered unreliable.

Solving this formula for t gives us t = -8267 ln(P) (note that the result will not be negative because P < 1, which means that the logarithm is also negative, giving us a positive result). We can now use this formula to approximate the age of an artifact if we know the percent of carbon-14 that it has remaining.

While the problems here involve relatively easy calculations, note that there is a lot more that goes into this in real life. For example, samples were taken from tree rings of very old trees to determine the amount of different types of carbon in the atmosphere over time. The information gained from these samples, along with other studies, helped scientists learn about carbon concentrations over time, which factors into the carbon dating process. All of this has an effect on the value of P, so for sake of ease, we will use given values of P.