1. **State the problem:** Solve the equation $$x^{\sqrt{x}} = 10$$ for $x$. 2. **Rewrite the equation:** Let $y = \sqrt{x}$. Then $x = y^2$. Substitute into the equation: $$x^{\sqrt{x}} = (y^2)^y = y^{2y} = 10$$ 3. **Simplify the expression:** We have $$y^{2y} = 10$$ 4. **Take the natural logarithm of both sides:** $$\ln(y^{2y}) = \ln(10)$$ 5. **Use logarithm power rule:** $$2y \ln(y) = \ln(10)$$ 6. **Rewrite the equation:** $$2y \ln(y) = \ln(10)$$ 7. **Isolate $y \ln(y)$:** $$y \ln(y) = \frac{\ln(10)}{2}$$ 8. **Solve for $y$ numerically:** This transcendental equation cannot be solved algebraically, so approximate $y$. 9. **Approximate $\ln(10) \approx 2.302585$:** $$y \ln(y) \approx 1.1512925$$ 10. **Try values for $y$:** - For $y=2$, $2 \ln(2) = 2 \times 0.6931 = 1.3862$ (too high) - For $y=1.7$, $1.7 \ln(1.7) = 1.7 \times 0.5306 = 0.902$ (too low) - For $y=1.9$, $1.9 \ln(1.9) = 1.9 \times 0.6419 = 1.2196$ (slightly high) - For $y=1.8$, $1.8 \ln(1.8) = 1.8 \times 0.5878 = 1.058$ (slightly low) 11. **Interpolate between 1.8 and 1.9:** Approximate $y \approx 1.85$. 12. **Calculate $x$:** $$x = y^2 \approx (1.85)^2 = 3.4225$$ 13. **Final answer:** $$\boxed{x \approx 3.42}$$