Radioactive decay is a first-order reaction. To estimate the age of the sample, we use the integrated rate law equation.

Step 1: Calculate the rate constant ($k$). For a first-order reaction, the relationship between half-life and rate constant is given by:

$$t_{1/2} = \frac{0.693}{k}$$

Given $t_{1/2} = 5730$ years.

$$k = \frac{0.693}{5730} = 1.21 \times 10^{-4} \text{ year}^{-1}$$

Step 2: Use the integrated rate law. The integrated rate law for a first-order reaction is:

$$t = \frac{2.303}{k} \log \frac{[R]_0}{[R]}$$

Given that the artifact has 80% of the $^{14}\text{C}$ found in a living tree, we can assume the initial concentration $[R]_0 = 100$ and the final concentration $[R] = 80$.

Step 3: Calculate the age ($t$). Substituting the values into the equation:

$$t = \frac{2.303}{1.21 \times 10^{-4}} \log \left(\frac{100}{80}\right)$$ $$t = \frac{2.303}{1.21 \times 10^{-4}} \log (1.25)$$ $$t = \frac{2.303}{1.21 \times 10^{-4}} (0.0969)$$ $$t = 1845 \text{ years}$$

The estimated age of the sample is 1845 years.