1. **State the problem:** Solve the equation $$\sqrt{5}^{2x-1} = \sqrt{5}^{-1}$$ for $x$. 2. **Recall the property of exponents:** If $a^m = a^n$ and $a > 0$, $a \neq 1$, then $m = n$. 3. **Rewrite the bases:** Note that $\sqrt{5} = 5^{1/2}$, so the equation becomes: $$\left(5^{1/2}\right)^{2x-1} = \left(5^{1/2}\right)^{-1}$$ 4. **Simplify the exponents:** Using the power of a power rule $\left(a^m\right)^n = a^{mn}$: $$5^{\frac{1}{2}(2x-1)} = 5^{\frac{1}{2}(-1)}$$ 5. **Simplify the exponents further:** $$5^{x - \frac{1}{2}} = 5^{-\frac{1}{2}}$$ 6. **Set the exponents equal:** Since the bases are equal and positive, we have: $$x - \frac{1}{2} = -\frac{1}{2}$$ 7. **Solve for $x$:** $$x = -\frac{1}{2} + \frac{1}{2} = 0$$ **Final answer:** $$x = 0$$