1. **State the problem:** Solve the inequality $\ln(x) > 4$. 2. **Recall the definition and properties:** The natural logarithm function $\ln(x)$ is defined for $x > 0$ and is the inverse of the exponential function $e^x$. 3. **Rewrite the inequality using the exponential function:** Since $\ln(x)$ is the inverse of $e^x$, the inequality $\ln(x) > 4$ is equivalent to $x > e^4$. 4. **Interpretation:** This means the values of $x$ for which the natural logarithm is greater than 4 are all $x$ greater than $e^4$. 5. **Final answer:** The solution set is $$x > e^4.$$