1. **State the problem:** Solve the equation $x^x = 36$ for $x$. 2. **Understand the equation:** The equation $x^x = 36$ is transcendental and cannot be solved using elementary algebraic methods. 3. **Use logarithms:** Take the natural logarithm on both sides: $$\ln(x^x) = \ln(36)$$ which simplifies to $$x \ln(x) = \ln(36)$$ 4. **Rewrite the equation:** Let $y = \ln(x)$, then $x = e^y$. Substitute into the equation: $$e^y \cdot y = \ln(36)$$ 5. **Use Lambert W function:** The equation can be rearranged as $$y e^y = \ln(36)$$ which implies $$y = W(\ln(36))$$ where $W$ is the Lambert W function. 6. **Find $x$:** Since $y = \ln(x)$, then $$x = e^y = e^{W(\ln(36))}$$ 7. **Approximate the value:** Numerically, $\ln(36) \approx 3.5835$ and $W(3.5835) \approx 1.3267$, so $$x \approx e^{1.3267} \approx 3.77$$ **Final answer:** $$x \approx 3.77$$