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

Our planet has experienced five major extinctions in the four billion year history of life. the first, 450 million years ago, occurred shortly after the evolution of the first land-based plants and 100 million years after the cambrian explosion. the second extinction occurred 350 million years ago, causing the formation of coal forests. next earth experienced two mass extinctions during the triassic period, between 250 and 200 million years ago. the fifth mass extinction occurred 65 million years ago, ending the reptilian dominance of the earth. according to richard leakey, the sixth mass extinction is happening right now. leakey suggests that we, the human race, are the cause. each year, at our hand, approximately 50,000 species vanish from earth. he believes that man is destroying earth at a rate comparable with the impact of a giant asteroid. leakey's statistics indicate that 50% of earth's species will become extinct within the next 100 years assuming leakey's hypothesis of a sixth mass extinction to be true, how will we expect the model to change? a) a sharp spike in the graph approximately 100 million years from now b) a dip in the graph, followed by a sharp spike about 100 million years from now c) a sharp spike in the graph immediately following the "0" location of the x axis d) a plateau following the "0" mark on the x axis, followed by a gradual rise to a new peak