To solve the equation \( \ln(1-x) = 2x \), we need to find \( x \).

1. **Start with the given equation:**
   \[
   \ln(1-x) = 2x
   \]

2. **Exponentiate both sides to eliminate the natural logarithm:**
   \[
   e^{\ln(1-x)} = e^{2x}
   \]

3. **Simplify using the property \( e^{\ln y} = y \):**
   \[
   1-x = e^{2x}
   \]

4. **Rearrange the equation to isolate the terms involving \( x \):**
   \[
   1 - x = e^{2x}
   \]
   \[
   x + e^{2x} = 1
   \]

5. **At this stage, we have a transcendental equation \( x + e^{2x} = 1 \). Solving it analytically is difficult, so we can look for numerical solutions.**

One common approach to find numerical solutions for transcendental equations is to use methods like the Newton-Raphson method, or use a graphing calculator or software to find the approximate solution.

**Using numerical methods (e.g., graphing):**

1. **Graph the functions \( y = x + e^{2x} \) and \( y = 1 \) to find their intersection point(s).**
2. **Alternatively, use a numerical solver to find the root(s) of the equation \( x + e^{2x} - 1 = 0 \).**

Let's use an iterative method (such as Newton-Raphson) to approximate the solution:

**Newton-Raphson Method:**

1. **Define the function \( f(x) \):**
   \[
   f(x) = x + e^{2x} - 1
   \]

2. **Compute its derivative \( f'(x) \):**
   \[
   f'(x) = 1 + 2e^{2x}
   \]

3. **Choose an initial guess \( x_0 \). Let's start with \( x_0 = 0 \).**

4. **Apply the Newton-Raphson iteration:**
   \[
   x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
   \]

**Iteration Steps:**

1. \( x_0 = 0 \)
   \[
   f(0) = 0 + e^{0} - 1 = 0
   \]
   \[
   f'(0) = 1 + 2e^{0} = 3
   \]
   \[
   x_1 = 0 - \frac{0}{3} = 0
   \]

Since \( x_0 = 0 \) is already a root after the first iteration (i.e., \( f(0) = 0 \)), we have:
\[
x = 0
\]

**Therefore, the solution to the equation \( \ln(1-x) = 2x \) is \( x = 0 \).**

Would you like more details or have any questions about the solution process?

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### Relative Questions:
1. How do we solve logarithmic equations in general?
2. What are transcendental equations and how do they differ from algebraic equations?
3. Can the Newton-Raphson method be applied to all types of equations?
4. What are some other numerical methods for solving transcendental equations?
5. How can graphing calculators help in solving complex equations?
6. What properties of logarithms are useful in solving equations?
7. How does exponentiation help in solving logarithmic equations?
8. What are the limitations of the Newton-Raphson method?

### Tip:
When solving equations involving logarithms, it's often helpful to remember the properties of logarithms and exponentials, such as \( e^{\ln(x)} = x \), to simplify and solve the equations effectively.

Learn how to solve the logarithmic equation ln(1-x) = 2x with detailed step-by-step instructions and numerical methods. This problem covers logarithmic and exponential equations, transcendental equations, and the Newton-Raphson method for numerical approximation. Suitable for advanced high school and college-level students.

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