1. **State the problem:** Solve the equation $25^{x-1} = 125^{2-x}$ for $x$. 2. **Recall the formula and rules:** We can express both sides with the same base to simplify the equation. Note that $25 = 5^2$ and $125 = 5^3$. 3. **Rewrite the equation using base 5:** $$25^{x-1} = (5^2)^{x-1} = 5^{2(x-1)}$$ $$125^{2-x} = (5^3)^{2-x} = 5^{3(2-x)}$$ 4. **Set the exponents equal since bases are the same:** $$2(x-1) = 3(2-x)$$ 5. **Expand both sides:** $$2x - 2 = 6 - 3x$$ 6. **Add $3x$ to both sides:** $$2x + 3x - 2 = 6 - 3x + 3x$$ $$5x - 2 = 6$$ 7. **Add 2 to both sides:** $$5x - \cancel{2} + \cancel{2} = 6 + 2$$ $$5x = 8$$ 8. **Divide both sides by 5:** $$\frac{5x}{\cancel{5}} = \frac{8}{\cancel{5}}$$ $$x = \frac{8}{5}$$ **Final answer:** $$x = \frac{8}{5}$$