Exponential and logarithmic functions in carbon 14 dating
The method of carbon dating makes use of the fact that all living organisms contain two isotopes of carbon, carbon12, denoted 12C (a stable isotope), and carbon14, denoted 14C (a radioactive isotope).The ratio of the amount of 14C to the amount of 12C is essentially constant (approximately 1/10,000).When an organism dies, the amount of 12C present remains unchanged, but the 14C decays at a rate proportional to the amount present with a halflife of approximately 5700 years.This change in the amount of 14C relative to the amount of 12C makes it possible to estimate the time at which the organism lived.
A fossil found in an archaeological dig was found to contain 20% of the original amount of 14C. I do not get the $0.693$ value, but perhaps my answer will help anyway.
If we assume Carbon14 decays continuously, then $$ C(t) = C_0e^, $$ where $C_0$ is the initial size of the sample. Since it takes 5,700 years for a sample to decay to half its size, we know $$ \frac C_0 = C_0e^, $$ which means $$ \frac = e^, $$ so the value of $C_0$ is irrelevant.
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