The task requires the student to use logarithms to solve an exponential equation in the realistic context of carbon dating, important in archaeology and geology, among other places.Students should be guided to recognize the use of the logarithm when the exponential function has the given base of $e$, as in this problem.Howstuffworks "How Carbon-14 Dating Works" radiocarbon WEB-info The method How accurate are Carbon-14 and other radioactive dating methods? Radiocarbon dating (usually referred to simply as carbon-14 dating) is a radiometric dating method.There are also trace amounts of the unstable radioisotope carbon-14 (14C) on Earth.Carbon-14 has a relatively short half-life of 5,730 years, meaning that the fraction of carbon-14 in a sample is halved over the course of 5,730 years due to radioactive decay to nitrogen-14.
$$ So either the answer is that ridiculously big number (9.17e7) or 30,476 years, being calculated with the equation I provided and the first equation in your answer, respectively.
The carbon-14 isotope would vanish from Earth's atmosphere in less than a million years were it not for the constant influx of cosmic rays interacting with molecules of nitrogen (N) into organic compounds during photosynthesis, the resulting fraction of the isotope 14C in the plant tissue will match the fraction of the isotope in the atmosphere.
After plants die or are consumed by other organisms, the incorporation of all carbon isotopes, including 14C, stops.
Oakley (1979) suggested its development meant an almost complete re-writing of the evolution and cultural emergence of the human species.
Desmond Clark (1979) wrote that were it not for radiocarbon dating, "we would still be foundering in a sea of imprecisions sometime bred of inspired guesswork but more often of imaginative speculation" (Clark, 1979:7).