1. **State the problem:** Solve the equation $2\ln(x) - 1 = 0$ for $x$. 2. **Recall the formula and rules:** The natural logarithm function $\ln(x)$ is defined for $x > 0$. To solve equations involving $\ln(x)$, isolate the logarithm and then exponentiate both sides to remove the logarithm. 3. **Isolate the logarithm:** $$2\ln(x) - 1 = 0 \implies 2\ln(x) = 1$$ 4. **Divide both sides by 2:** $$\ln(x) = \frac{1}{2}$$ 5. **Exponentiate both sides to solve for $x$:** $$x = e^{\frac{1}{2}}$$ 6. **Simplify the expression:** $$x = \sqrt{e}$$ 7. **Check the domain:** Since $x = \sqrt{e} > 0$, it is valid. **Final answer:** $$x = \sqrt{e}$$