1. **State the problem:** Solve the equation $0.1^{2x-1} = 100$ for $x$. 2. **Recall the formula and rules:** We use the property of exponents and logarithms. Since $0.1 = 10^{-1}$ and $100 = 10^2$, rewrite the equation with the same base. 3. **Rewrite the bases:** $$0.1^{2x-1} = (10^{-1})^{2x-1} = 10^{-1(2x-1)} = 10^{-2x+1}$$ 4. **Set the exponents equal:** Since $10^{-2x+1} = 10^2$, the exponents must be equal: $$-2x + 1 = 2$$ 5. **Solve for $x$:** $$-2x + 1 = 2$$ $$-2x = 2 - 1$$ $$-2x = 1$$ $$x = \frac{1}{-2} = -\frac{1}{2}$$ 6. **Final answer:** $$x = -\frac{1}{2}$$