The equation \(F=e^{-k t}\) describes the fraction \(F\) of an original isotope
remaining after a period of \(t\) years; the exponent is negative because it
refers to a decrease over time. The constant \(k\) provides a measure of how
rapidly the original isotope decays. For the decay of carbon- 14 to nitrogen-
\(14, k=0.00012097\). To find \(t\), rearrange the equation by following these
steps: (a) Take the natural logarithm of both sides of the equation: \(\ln
(F)=\) In(e \({ }^{-k t}\) ). Rewrite the right side of this equation by applying
the following rule: \(\ln \left(e^{x}\right)=x \ln (e)\). (b) Since \(\ln (e)=1\),
simplify the equation. (c) Now solve for \(t\) and write the equation in the
form
