Solve Exponential 3A80F9

1. **Problem Statement:** Solve for $x$ in the equation $$x^{x^x} = 10$$. 2. **Understanding the problem:** This is an exponential equation where the exponent itself is an exponential expression. 3. **Step 1: Let’s set $y = x^x$.** Then the equation becomes $$x^y = 10$$. 4. **Step 2: Express $x^y = 10$ as $$y \log x = \log 10$$.** Since $\log 10 = 1$ (assuming base 10 logarithm), we have $$y \log x = 1$$. 5. **Step 3: Recall that $y = x^x$, so substitute back:** $$x^x \log x = 1$$. 6. **Step 4: This is a transcendental equation and cannot be solved algebraically in closed form.** We can try to find an approximate solution by trial or numerical methods. 7. **Step 5: Try $x=1.5$:** $$x^x = 1.5^{1.5} \approx 1.837$$ $$x^x \log x = 1.837 \times \log 1.5 \approx 1.837 \times 0.1761 = 0.323 < 1$$ 8. **Step 6: Try $x=2$:** $$x^x = 2^2 = 4$$ $$x^x \log x = 4 \times \log 2 = 4 \times 0.3010 = 1.204 > 1$$ 9. **Step 7: Since at $x=1.5$ value is less than 1 and at $x=2$ value is greater than 1, the solution lies between 1.5 and 2.** 10. **Step 8: Try $x=1.7$:** $$x^x = 1.7^{1.7} \approx 2.35$$ $$x^x \log x = 2.35 \times \log 1.7 = 2.35 \times 0.2304 = 0.541 < 1$$ 11. **Step 9: Try $x=1.9$:** $$x^x = 1.9^{1.9} \approx 3.45$$ $$x^x \log x = 3.45 \times \log 1.9 = 3.45 \times 0.2788 = 0.961 < 1$$ 12. **Step 10: Try $x=1.95$:** $$x^x = 1.95^{1.95} \approx 3.75$$ $$x^x \log x = 3.75 \times \log 1.95 = 3.75 \times 0.2907 = 1.09 > 1$$ 13. **Step 11: The solution is between 1.9 and 1.95.** 14. **Step 12: Approximate solution is about $x \approx 1.92$.** 15. **Final answer:** $$\boxed{x \approx 1.92}$$