1. **State the problem:** Solve the equation $$\frac{\ln(-2x)}{5} - 5x = 0$$ for $x$. 2. **Rewrite the equation:** Multiply both sides by 5 to clear the denominator: $$\ln(-2x) - 25x = 0$$ 3. **Isolate the logarithm:** $$\ln(-2x) = 25x$$ 4. **Domain consideration:** The argument of the logarithm must be positive: $$-2x > 0 \implies x < 0$$ 5. **Rewrite the equation using exponentiation:** Exponentiate both sides to remove the logarithm: $$-2x = e^{25x}$$ 6. **Rewrite as:** $$-2x = e^{25x}$$ 7. **Check for solutions:** This transcendental equation cannot be solved algebraically in closed form. We analyze the function: $$f(x) = -2x - e^{25x}$$ 8. **Check values for $x<0$:** - At $x=0$, $f(0) = 0 - 1 = -1$ (negative) - At $x=-0.1$, $f(-0.1) = -2(-0.1) - e^{-2.5} = 0.2 - 0.0821 = 0.1179$ (positive) Since $f(x)$ changes from positive to negative between $-0.1$ and $0$, there is a root in that interval. 9. **Approximate solution:** Using numerical methods (e.g., bisection), the root is approximately: $$x \approx -0.044$$ **Final answer:** $$x \approx -0.044$$