Question: A logarithm is the power to which a base is raised to produce a

given number x. For example, if t…

A logarithm 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)−k |

t = ln(F)ln(e−k) |

t = ln(F)−k∗ln(e) |

t = ln(F)−k ( Correctanswer) |

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

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