1. **State the problem:** Solve the exponential equation $$5^{x^x} = 5^x + 26$$. 2. **Analyze the equation:** The equation involves an exponential expression with a variable exponent inside another exponent, which is unusual. We want to find values of $x$ that satisfy this. 3. **Try to find integer solutions by testing values:** - For $x=1$: $$5^{1^1} = 5^1 + 26 \Rightarrow 5^1 = 5 + 26 \Rightarrow 5 = 31$$ (False) - For $x=2$: $$5^{2^2} = 5^2 + 26 \Rightarrow 5^4 = 25 + 26 \Rightarrow 625 = 51$$ (False) - For $x=3$: $$5^{3^3} = 5^3 + 26 \Rightarrow 5^{27} = 125 + 26 \Rightarrow \text{very large} = 151$$ (False) 4. **Try $x=0$:** $$5^{0^0} = 5^0 + 26$$ Note: $0^0$ is indeterminate, so discard. 5. **Try $x= -1$:** $$5^{(-1)^{-1}} = 5^{-1} + 26 \Rightarrow 5^{-1} = \frac{1}{5}$$ $$5^{-1} = \frac{1}{5} + 26 \Rightarrow \frac{1}{5} = 26.2$$ (False) 6. **Try $x= 2$ again carefully:** $$5^{2^2} = 5^4 = 625$$ $$5^2 + 26 = 25 + 26 = 51$$ Not equal. 7. **Try $x= 1.5$:** Calculate $x^x = 1.5^{1.5} = e^{1.5 \ln 1.5} \approx e^{1.5 \times 0.4055} = e^{0.608} \approx 1.837$ Then: $$5^{1.837} \approx e^{1.837 \ln 5} = e^{1.837 \times 1.609} = e^{2.956} \approx 19.2$$ $$5^{1.5} + 26 = 5^{1.5} + 26 = e^{1.5 \times 1.609} + 26 = e^{2.414} + 26 \approx 11.18 + 26 = 37.18$$ Not equal. 8. **Try $x= 1.7$:** $$x^x = 1.7^{1.7} = e^{1.7 \ln 1.7} = e^{1.7 \times 0.5306} = e^{0.902} = 2.465$$ $$5^{2.465} = e^{2.465 \times 1.609} = e^{3.967} = 52.8$$ $$5^{1.7} + 26 = e^{1.7 \times 1.609} + 26 = e^{2.735} + 26 = 15.4 + 26 = 41.4$$ Not equal. 9. **Try $x= 1.6$:** $$x^x = 1.6^{1.6} = e^{1.6 \ln 1.6} = e^{1.6 \times 0.4700} = e^{0.752} = 2.12$$ $$5^{2.12} = e^{2.12 \times 1.609} = e^{3.41} = 30.2$$ $$5^{1.6} + 26 = e^{1.6 \times 1.609} + 26 = e^{2.574} + 26 = 13.1 + 26 = 39.1$$ Not equal. 10. **Try $x= 1.55$:** $$x^x = 1.55^{1.55} = e^{1.55 \ln 1.55} = e^{1.55 \times 0.4383} = e^{0.679} = 1.97$$ $$5^{1.97} = e^{1.97 \times 1.609} = e^{3.17} = 23.8$$ $$5^{1.55} + 26 = e^{1.55 \times 1.609} + 26 = e^{2.49} + 26 = 12.1 + 26 = 38.1$$ Not equal. 11. **Try $x= 1.4$:** $$x^x = 1.4^{1.4} = e^{1.4 \ln 1.4} = e^{1.4 \times 0.3365} = e^{0.471} = 1.60$$ $$5^{1.60} = e^{1.60 \times 1.609} = e^{2.574} = 13.1$$ $$5^{1.4} + 26 = e^{1.4 \times 1.609} + 26 = e^{2.25} + 26 = 9.5 + 26 = 35.5$$ Not equal. 12. **Try $x= 1.3$:** $$x^x = 1.3^{1.3} = e^{1.3 \ln 1.3} = e^{1.3 \times 0.2624} = e^{0.341} = 1.41$$ $$5^{1.41} = e^{1.41 \times 1.609} = e^{2.27} = 9.7$$ $$5^{1.3} + 26 = e^{1.3 \times 1.609} + 26 = e^{2.09} + 26 = 8.1 + 26 = 34.1$$ Not equal. 13. **Try $x= 1$:** $$5^{1^1} = 5^1 = 5$$ $$5^1 + 26 = 5 + 26 = 31$$ Not equal. 14. **Try $x= 0$:** $$5^{0^0}$$ is undefined, so discard. 15. **Try $x= 3$:** $$5^{3^3} = 5^{27}$$ is huge, while $$5^3 + 26 = 125 + 26 = 151$$ Not equal. 16. **Try $x= 2.5$:** $$x^x = 2.5^{2.5} = e^{2.5 \ln 2.5} = e^{2.5 \times 0.9163} = e^{2.29} = 9.87$$ $$5^{9.87} = e^{9.87 \times 1.609} = e^{15.88} = 7.9 \times 10^6$$ $$5^{2.5} + 26 = e^{2.5 \times 1.609} + 26 = e^{4.02} + 26 = 55.7 + 26 = 81.7$$ Not equal. 17. **Conclusion:** The only integer solution that works is $x=2$ if we check the original equation carefully: $$5^{2^2} = 5^4 = 625$$ $$5^2 + 26 = 25 + 26 = 51$$ Not equal, so no integer solution. 18. **Try $x= 1.7$ again:** $$5^{1.7^{1.7}} = 5^{2.465} = 52.8$$ $$5^{1.7} + 26 = 15.4 + 26 = 41.4$$ Close but no. 19. **Try $x= 1.8$:** $$x^x = 1.8^{1.8} = e^{1.8 \ln 1.8} = e^{1.8 \times 0.5878} = e^{1.058} = 2.88$$ $$5^{2.88} = e^{2.88 \times 1.609} = e^{4.63} = 102.5$$ $$5^{1.8} + 26 = e^{1.8 \times 1.609} + 26 = e^{2.9} + 26 = 18.2 + 26 = 44.2$$ Not equal. 20. **Try $x= 1.9$:** $$x^x = 1.9^{1.9} = e^{1.9 \ln 1.9} = e^{1.9 \times 0.6419} = e^{1.22} = 3.39$$ $$5^{3.39} = e^{3.39 \times 1.609} = e^{5.46} = 235.5$$ $$5^{1.9} + 26 = e^{1.9 \times 1.609} + 26 = e^{3.06} + 26 = 21.3 + 26 = 47.3$$ Not equal. 21. **Try $x= 1.65$:** $$x^x = 1.65^{1.65} = e^{1.65 \ln 1.65} = e^{1.65 \times 0.5008} = e^{0.826} = 2.28$$ $$5^{2.28} = e^{2.28 \times 1.609} = e^{3.67} = 39.3$$ $$5^{1.65} + 26 = e^{1.65 \times 1.609} + 26 = e^{2.65} + 26 = 14.1 + 26 = 40.1$$ Close. 22. **Try $x= 1.63$:** $$x^x = 1.63^{1.63} = e^{1.63 \ln 1.63} = e^{1.63 \times 0.489} = e^{0.797} = 2.22$$ $$5^{2.22} = e^{2.22 \times 1.609} = e^{3.57} = 35.6$$ $$5^{1.63} + 26 = e^{1.63 \times 1.609} + 26 = e^{2.62} + 26 = 13.7 + 26 = 39.7$$ Not equal. 23. **Try $x= 1.67$:** $$x^x = 1.67^{1.67} = e^{1.67 \ln 1.67} = e^{1.67 \times 0.512} = e^{0.855} = 2.35$$ $$5^{2.35} = e^{2.35 \times 1.609} = e^{3.78} = 43.9$$ $$5^{1.67} + 26 = e^{1.67 \times 1.609} + 26 = e^{2.69} + 26 = 14.7 + 26 = 40.7$$ Not equal. 24. **Summary:** The equation has no simple closed form solution. The approximate solution is near $x \approx 1.65$ where both sides are close. **Final answer:** The solution to $$5^{x^x} = 5^x + 26$$ is approximately $$x \approx 1.65$$. This can be verified numerically or solved using numerical methods like Newton-Raphson.