1. **State the problem:** Solve the equation $$\frac{1}{1 - e^x} = 5$$ for $$x$$. 2. **Rewrite the equation:** Multiply both sides by $$1 - e^x$$ to clear the denominator: $$1 = 5(1 - e^x)$$ 3. **Distribute the 5:** $$1 = 5 - 5e^x$$ 4. **Isolate the exponential term:** Subtract 5 from both sides: $$1 - 5 = -5e^x$$ $$-4 = -5e^x$$ 5. **Divide both sides by -5:** $$\frac{-4}{-5} = e^x$$ Intermediate step showing cancellation: $$\frac{\cancel{-4}}{\cancel{-5}} = e^x$$ Simplifies to: $$\frac{4}{5} = e^x$$ 6. **Take the natural logarithm of both sides:** $$\ln\left(e^x\right) = \ln\left(\frac{4}{5}\right)$$ 7. **Use the property $$\ln(e^x) = x$$:** $$x = \ln\left(\frac{4}{5}\right)$$ 8. **Final answer:** $$x = \ln\left(\frac{4}{5}\right) \approx -0.223$$ This is the solution to the equation.