1. **State the problem:** Solve the equation $3^x - 2 = 5^{x+1}$ for $x$. 2. **Rewrite the equation:** The equation is $3^x - 2 = 5^{x+1}$. 3. **Express $5^{x+1}$:** Use the property of exponents: $5^{x+1} = 5^x \cdot 5^1 = 5 \cdot 5^x$. 4. **Rewrite the equation:** $$3^x - 2 = 5 \cdot 5^x$$ 5. **Isolate terms:** Move all terms to one side: $$3^x - 5 \cdot 5^x = 2$$ 6. **Rewrite the equation:** $$3^x - 5 \cdot 5^x - 2 = 0$$ 7. **Use substitution:** Let $a = 3^x$ and $b = 5^x$. 8. **Note:** Since $a$ and $b$ are exponential functions with different bases, direct substitution is complicated. Instead, try to solve numerically or by taking logarithms. 9. **Rewrite original equation:** $$3^x - 2 = 5^{x+1}$$ 10. **Take natural logarithm on both sides:** $$\ln(3^x - 2) = \ln(5^{x+1})$$ 11. **Simplify right side:** $$\ln(5^{x+1}) = (x+1) \ln 5$$ 12. **Note:** The left side is $\ln(3^x - 2)$ which is not easily simplified. 13. **Try numerical approach:** Test values of $x$ to find approximate solution. - For $x=1$: $3^1 - 2 = 1$, $5^{1+1} = 25$, not equal. - For $x=2$: $3^2 - 2 = 7$, $5^{3} = 125$, not equal. - For $x=0$: $3^0 - 2 = 1 - 2 = -1$, $5^{1} = 5$, no. 14. **Try $x=-1$:** $$3^{-1} - 2 = \frac{1}{3} - 2 = -\frac{5}{3}$$ $$5^{0} = 1$$ No. 15. **Try $x=3$:** $$3^3 - 2 = 27 - 2 = 25$$ $$5^{4} = 625$$ No. 16. **Try $x=0.5$:** $$3^{0.5} - 2 = \sqrt{3} - 2 \approx 1.732 - 2 = -0.268$$ $$5^{1.5} = 5^{1} \cdot 5^{0.5} = 5 \cdot \sqrt{5} \approx 5 \cdot 2.236 = 11.18$$ No. 17. **Try $x= -0.5$:** $$3^{-0.5} - 2 = \frac{1}{\sqrt{3}} - 2 \approx 0.577 - 2 = -1.423$$ $$5^{0.5} = \sqrt{5} \approx 2.236$$ No. 18. **Try $x=4$:** $$3^4 - 2 = 81 - 2 = 79$$ $$5^{5} = 3125$$ No. 19. **Observation:** Left side grows slower than right side for positive $x$. 20. **Try to solve by rewriting:** $$3^x - 2 = 5^{x+1}$$ $$3^x - 2 = 5 \cdot 5^x$$ $$3^x - 5 \cdot 5^x = 2$$ 21. **Divide both sides by $5^x$:** $$\frac{3^x}{5^x} - 5 = \frac{2}{5^x}$$ 22. **Rewrite:** $$\left(\frac{3}{5}\right)^x - 5 = \frac{2}{5^x}$$ 23. **Let $y = \left(\frac{3}{5}\right)^x$ and $z = 5^{-x}$:** Note that $z = \frac{1}{5^x}$. 24. **Rewrite equation:** $$y - 5 = 2z$$ 25. **But $z = \frac{1}{5^x} = \left(\frac{1}{5}\right)^x$ and $y = \left(\frac{3}{5}\right)^x$. 26. **Since $\frac{3}{5} > \frac{1}{5}$, $y$ decreases slower than $z$ as $x$ increases. Try to find $x$ numerically.** 27. **Try $x=0$:** $$y = 1, z = 1$$ $$1 - 5 = 2(1) \Rightarrow -4 = 2$$ No. 28. **Try $x=-1$:** $$y = \left(\frac{3}{5}\right)^{-1} = \frac{5}{3} \approx 1.666$$ $$z = \left(\frac{1}{5}\right)^{-1} = 5$$ $$1.666 - 5 = 2 \times 5 \Rightarrow -3.334 = 10$$ No. 29. **Try $x=1$:** $$y = \frac{3}{5} = 0.6$$ $$z = \frac{1}{5} = 0.2$$ $$0.6 - 5 = 2 \times 0.2 \Rightarrow -4.4 = 0.4$$ No. 30. **Try $x=2$:** $$y = \left(\frac{3}{5}\right)^2 = \frac{9}{25} = 0.36$$ $$z = \left(\frac{1}{5}\right)^2 = \frac{1}{25} = 0.04$$ $$0.36 - 5 = 2 \times 0.04 \Rightarrow -4.64 = 0.08$$ No. 31. **Try $x=-2$:** $$y = \left(\frac{3}{5}\right)^{-2} = \left(\frac{5}{3}\right)^2 = \frac{25}{9} \approx 2.778$$ $$z = \left(\frac{1}{5}\right)^{-2} = 25$$ $$2.778 - 5 = 2 \times 25 \Rightarrow -2.222 = 50$$ No. 32. **Conclusion:** No real solution satisfies the equation because the left side is always less than the right side for all tested values. 33. **Check for extraneous solutions:** The original equation involves $3^x - 2$ which must be positive to take logarithms. 34. **Final answer:** The equation $3^x - 2 = 5^{x+1}$ has no real solution. **Answer:** No real solution exists for $x$.