$$ 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.

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.

Working with several collaboraters, Libby established the natural occurrence of radiocarbon by detecting its radioactivity in methane from the Baltimore sewer.

In contrast, methane made from petroleum products had no measurable radioactivity.