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:

[latex]A\left(t\right)=A_0e^{-kt}[/latex]

where

[latex]A(t)[/latex] = mass of element remaining after t years

[latex]A_0[/latex] = original mass of element

[latex]k[/latex] = rate of decay

[latex]t[/latex] = 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 [latex]A(t)[/latex] and [latex]A_0[/latex] as follows:

[latex]A\left(t\right)=A_0e^{-kt}[/latex]

[latex]\frac{1}{2}=(1)e^{-k\left(5730\right)}[/latex]

[latex]1n\left(\frac{1}{2}\right)=1n\left(e^{-k\left(5730\right)}\right)[/latex]

[latex]1n\left(2^{-1}\right)=\left(-5730k\right)1n\left(e\right)[/latex]

[latex]-1n\left(2\right)=-5730k\left(1\right)[/latex]

[latex]k\approx1.21\times10^{-4}[/latex]

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 [latex]A(t)=100e^{\large{-(0.000121)t}}[/latex]
Substitute 20 grams for A(t) [latex]20=100e^{\large{-\left(0.000121\right)t}}[/latex]
Divide both sides by 100 [latex]0.20=e^{\large{-\left(0.000121\right)t}}[/latex]
Change to logarithmic form [latex]1n\left(0.20\right)=-\left(0.000121\right)t[/latex]
Divide both sides by -0.000121 [latex]t={\large\frac{1n\left(0.20\right)}{-0.000121}}[/latex]
Solve [latex]t\approx13,301[/latex] 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.

[latex]A\left(t\right)=100e^{-\left(0.000121\right)t}[/latex]

[latex]A\left(8000\right)=100e^{-\left(0.000121\right)\left(8000\right)}\approx38[/latex] 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.