1. **Problem statement:** Find the area of the region bounded by the curve $xy=4$, the x-axis, and the vertical lines $x=1$ and $x=3$. 2. **Rewrite the curve:** From $xy=4$, express $y$ as a function of $x$: $$y=\frac{4}{x}$$ 3. **Area under the curve:** The area bounded by the curve, x-axis, and vertical lines is given by the definite integral: $$\text{Area} = \int_1^3 \frac{4}{x} \, dx$$ 4. **Integral formula:** Recall that $$\int \frac{1}{x} \, dx = \ln|x| + C$$ 5. **Evaluate the integral:** $$\int_1^3 \frac{4}{x} \, dx = 4 \int_1^3 \frac{1}{x} \, dx = 4 [\ln|x|]_1^3 = 4 (\ln 3 - \ln 1)$$ 6. **Simplify:** Since $\ln 1 = 0$, $$4 (\ln 3 - 0) = 4 \ln 3$$ 7. **Final answer:** The area is $$\boxed{4 \ln 3}$$ This corresponds to option (b).