Alogarithm is the power to which a base is raised to produce a given number x. for example, if the base is 10 and x = 100, the logarithm of 100 equals 2 (because 102 = 100). a natural logarithm (ln) is the logarithm of a number x to the base e, where e is about 2.718. natural logarithms are useful in calculating rates of some natural processes, such as radioactive decay. the equation f = e –kt 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. you can rearrange this equation to solve for t by following these steps: take the natural logarithm of both sides of the equation: ln(f) = ln(e –kt ). rewrite the right side of this equation by applying the following rule: ln(ex ) = x ln(e). since ln(e) = 1, simplify the equation. now solve for t in the form "t = "t = log10(f)−kt = ln(f)ln(e−k)t = ln(f)−k∗ln(e)t = ln(f)−k ( correct answer)the equation t = ln(f)−k enables you to calculate the age (t) of an ancient seed based on the fraction of carbon-14 remaining (f). the constant k is the rate at which carbon-14 decays to nitrogen-14, leading to a decline in carbon-14 over time