The equation shown is:

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

Steps to Solve Algebraically:

1. Simplify the equation: Multiply through by $1 - e^{-x}$ to eliminate the fraction: $1 = 5(1 - e^{-x})$

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

3. Rearrange terms: Isolate $e^{-x}$: $5e^{-x} = 5 - 1$ $5e^{-x} = 4$

4. Solve for $e^{-x}$: Divide by 5: $e^{-x} = \frac{4}{5} = 0.8$

5. Take the natural logarithm (ln): Apply $\ln$ to both sides: $-x = \ln(0.8)$

6. Solve for $x$: $x = -\ln(0.8)$

Approximation:

Use a calculator to evaluate: $x = -\ln(0.8) \approx 0.223$

The provided answer $x = 0.223$ is correct.

Would you like further explanation or clarifications? Here are some related questions to deepen your understanding:

1. What properties of natural logarithms are important for solving exponential equations like this one?
2. How does changing the constant (e.g., from 5 to another number) affect the solution process?
3. Why is it necessary to isolate the exponential term $e^{-x}$ before taking the natural logarithm?
4. Can this equation be solved graphically, and how would the solution compare to the algebraic method?
5. How can you verify the solution by substituting $x = 0.223$ back into the original equation?

Tip: Always check your solution by plugging it back into the original equation to confirm it satisfies the problem!