To solve the given exponential equation:

$\frac{1}{1 - e^{-x}} = 5$

Step-by-Step Solution:

1. Simplify the equation: Multiply both sides by $1 - e^{-x}$ (assuming $1 - e^{-x} \neq 0$): $1 = 5(1 - e^{-x})$

2. Expand the terms: $1 = 5 - 5e^{-x}$

3. Isolate the exponential term: $5e^{-x} = 5 - 1$ $5e^{-x} = 4$

4. Solve for $e^{-x}$: $e^{-x} = \frac{4}{5}$

5. Take the natural logarithm (ln) on both sides: $-x = \ln\left(\frac{4}{5}\right)$

6. Solve for $x$: $x = -\ln\left(\frac{4}{5}\right)$

7. Simplify further using logarithmic properties: $x = \ln\left(\frac{5}{4}\right)$

Final Approximation:

Use a calculator to compute: $x \approx \ln\left(1.25\right) \approx 0.223$

Final Answer:

$x \approx 0.223$

Would you like me to go over any steps in detail? Here are related questions you may explore:

1. What are the properties of logarithms used in this solution?
2. How does the natural logarithm function behave for values less than 1?
3. Can this equation be solved graphically? If so, how?
4. How would the solution change if the equation involved $e^x$ instead of $e^{-x}$?
5. Why do we multiply through by $1 - e^{-x}$ initially?

Tip: Always check the domain of your functions (e.g., $1 - e^{-x} \neq 0$) to avoid invalid solutions!