Carbon dating problem
Fortunately, we do have an organic object that tracks carbon in the atmosphere on a yearly basis: tree rings.
Trees maintain carbon 14 equilibrium in their growth rings—and trees produce a ring for every year they are alive.
Now, take the logarithm of both sides to get $$ -0.693 = -5700k, $$ from which we can derive $$ k \approx 1.22 \cdot 10^.
Other organic data sets examined have included varves (layers in sedimentary rock which were laid down annually and contain organic materials, deep ocean corals, speleothems (cave deposits), and volcanic tephras; but there are problems with each of these methods.Although we don't have any 50,000-year-old trees, we do have overlapping tree ring sets back to 12,594 years.So, in other words, we have a pretty solid way to calibrate raw radiocarbon dates for the most recent 12,594 years of our planet's past.A fossil found in an archaeological dig was found to contain 20% of the original amount of 14C. I do not get the $-0.693$ value, but perhaps my answer will help anyway.
If we assume Carbon-14 decays continuously, then $$ 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.
A child mummy is found high in the Andes and the archaeologist says the child lived more than 2,000 years ago.