Math Problem Statement

Solve the exponential equation algebraically: \( \frac{1}{1 - e^{-x}} = 5 \). Approximate the result to three decimal places.

Solution

The equation shown is:

11ex=5\frac{1}{1 - e^{-x}} = 5

Steps to Solve Algebraically:

  1. Simplify the equation: Multiply through by 1ex1 - e^{-x} to eliminate the fraction: 1=5(1ex)1 = 5(1 - e^{-x})

  2. Distribute the 5: 1=55ex1 = 5 - 5e^{-x}

  3. Rearrange terms: Isolate exe^{-x}: 5ex=515e^{-x} = 5 - 1 5ex=45e^{-x} = 4

  4. Solve for exe^{-x}: Divide by 5: ex=45=0.8e^{-x} = \frac{4}{5} = 0.8

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

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

Approximation:

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

The provided answer x=0.223x = 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 exe^{-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=0.223x = 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!

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Logarithms

Formulas

Natural Logarithm: \(\ln(a)\)
Exponential equation manipulation

Theorems

Logarithmic Properties
Inverse Relationship between Exponentials and Logarithms

Suitable Grade Level

College-Level Algebra