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Study Guides > College Algebra CoRequisite Course

Putting It Together: Exponential and Logarithmic Equations and Models

At the start of this module, you were assigned the task of analyzing a fossilized bone to determine its age. To make that estimate, you need to model for the decay rate of carbon-14. The decay of a radioactive element is an exponential function of the form:

A(t)=A0e−ktA\left(t\right)=A_0e^{-kt}

where

A(t)A(t) = mass of element remaining after t years

A0A_0 = original mass of element

kk = rate of decay

tt = time in years

So to create a model for the decay function of carbon-14, assume for simplicity that the sample you started with had a mass of 1g. We know that half the starting mass of the sample will remain after one half-life which is 5,730 years. We can substitute these values for A(t)A(t) and A0A_0 as follows:

A(t)=A0e−ktA\left(t\right)=A_0e^{-kt}

12=(1)e−k(5730)\frac{1}{2}=(1)e^{-k\left(5730\right)}

1n(12)=1n(e−k(5730))1n\left(\frac{1}{2}\right)=1n\left(e^{-k\left(5730\right)}\right)

1n(2−1)=(−5730k)1n(e)1n\left(2^{-1}\right)=\left(-5730k\right)1n\left(e\right)

−1n(2)=−5730k(1)-1n\left(2\right)=-5730k\left(1\right)

k≈1.21×10−4k\approx1.21\times10^{-4}

Now you know the decay rate so you can write the equation for the exponential decay of carbon-14 and you can represent it as a graph. graph shows percentage of decay over time starting at 100% remaining to 0% over roughly 47500 years. The next step is to evaluate the function for a given mass. Assume a starting mass of 100 grams and that there are 20 grams remaining. Substitute these values into the model in the following way:
Write the equation A(t)=100e−(0.000121)tA(t)=100e^{\large{-(0.000121)t}}
Substitute 20 grams for A(t) 20=100e−(0.000121)t20=100e^{\large{-\left(0.000121\right)t}}
Divide both sides by 100 0.20=e−(0.000121)t0.20=e^{\large{-\left(0.000121\right)t}}
Change to logarithmic form 1n(0.20)=−(0.000121)t1n\left(0.20\right)=-\left(0.000121\right)t
Divide both sides by -0.000121 t=1n(0.20)−0.000121t={\large\frac{1n\left(0.20\right)}{-0.000121}}
Solve t≈13,301t\approx13,301 years
Now you know that it would take 13,301 years for a 100-gram sample of carbon-14 to decay to the point that only 20 grams are left. Confirm that this number makes sense by looking at the graph. You can also determine the amount of a 100-gram sample that would remain after a given number of years such as 8,000.  To do this, substitute the number of years into the function and evaluate.

A(t)=100e−(0.000121)tA\left(t\right)=100e^{-\left(0.000121\right)t}

A(8000)=100e−(0.000121)(8000)≈38A\left(8000\right)=100e^{-\left(0.000121\right)\left(8000\right)}\approx38 grams

About 38 grams would remain after 8,000 years. Understanding exponential functions helps scientists better understand radioactive decay and provides insights into past civilizations and species.

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