Let's break down and solve this problem step by step.
### Step 1: Understand the given information
Importantly, we know:
- The decay formula for carbon-14 is given by the relationship \( R = \left( \frac{1}{2} \right)^{\frac{t}{5730}} \), where \( R \) is the ratio of carbon-14 to carbon-12, \( t \) is the time in years, and 5730 years is the half-life of carbon-14.
- The ratio \( R = \frac{1}{8^{16}} \).
### Step 2: Setting up the equation
Given \( R = \left( \frac{1}{2} \right)^{\frac{t}{5730}} \) and \( R = \frac{1}{8^{16}} \), we equate the two expressions for \( R \):
[tex]\[ \left( \frac{1}{2} \right)^{\frac{t}{5730}} = \frac{1}{8^{16}} \][/tex]
### Step 3: Transform the right-hand side expression for simplicity
We rewrite \(\frac{1}{8^{16}}\) to match the left-hand side form:
[tex]\[ 8 = 2^3 \implies 8^{16} = (2^3)^{16} = 2^{48} \implies \frac{1}{8^{16}} = \frac{1}{2^{48}} = 2^{-48} \][/tex]
So the equation becomes:
[tex]\[ \left( \frac{1}{2} \right)^{\frac{t}{5730}} = 2^{-48} \][/tex]
### Step 4: Solve for \( t \) using logarithms
We take the natural logarithm (\(\ln\)) of both sides to solve for \( t \):
[tex]\[ \ln \left( \left( \frac{1}{2} \right)^{\frac{t}{5730}} \right) = \ln (2^{-48}) \][/tex]
Using the logarithm property \(\ln (a^b) = b \ln (a)\), we get:
[tex]\[ \frac{t}{5730} \ln \left( \frac{1}{2} \right) = \ln (2^{-48}) \][/tex]
Since \(\ln \left( \frac{1}{2} \right) = -\ln (2)\) and \(\ln (2^{-48}) = -48 \ln (2)\), the equation simplifies to:
[tex]\[ \frac{t}{5730} (-\ln (2)) = -48 \ln (2) \][/tex]
### Step 5: Isolate \( t \)
We can now solve for \( t \):
[tex]\[ t = \frac{-48 \ln (2) \cdot 5730}{-\ln (2)} \][/tex]
The \(\ln (2)\) cancels out:
[tex]\[ t = 48 \cdot 5730 \][/tex]
### Conclusion
Thus, the age \( t \) of the piece of wood is:
[tex]\[ t = 275040 \][/tex]
Rewriting this in the form given in the problem:
[tex]\[
t = (-8223) \frac{\ln \left(2^{-48}\right)}{\ln (e)} = 275040 \text{ years}
\][/tex]
This completes our detailed step-by-step solution for determining the age of the piece of wood given the provided ratio of carbon-14 to carbon-12.
