To find the range of the function \( f(x) = \frac{1}{\ln(x) - 1} \), we need to consider the domain of the logarithmic function and the behavior of the expression as \( \ln(x) \) varies.

### Step 1: Identify the domain of the function
The logarithmic function \( \ln(x) \) is defined for \( x > 0 \), and \( \ln(x) = 0 \) when \( x = 1 \). Additionally, \( \ln(x) \) approaches \( -\infty \) as \( x \) approaches \( 0^+ \), and \( \ln(x) \) approaches \( +\infty \) as \( x \) increases.

However, for the function \( f(x) \) to be defined, the denominator \( \ln(x) - 1 \) must not be zero:
\[
\ln(x) - 1 \neq 0 \Rightarrow \ln(x) \neq 1 \Rightarrow x \neq e
\]
Thus, the domain of \( f(x) \) is \( x > 0 \) except \( x = e \).

### Step 2: Analyze the behavior of the function
- **As \( x \) approaches 0 from the right (\( x \to 0^+ \)):** \( \ln(x) \) approaches \( -\infty \), so \( \ln(x) - 1 \) also approaches \( -\infty \). Therefore, \( f(x) = \frac{1}{\ln(x) - 1} \) approaches 0 from the negative side.

- **As \( x \) increases towards \( e \) from the left (\( x \to e^- \)):** \( \ln(x) \) approaches 1 from below, making \( \ln(x) - 1 \) approach 0 from the negative side, so \( f(x) \) approaches \( -\infty \).

- **As \( x \) increases beyond \( e \) (\( x > e \)):** \( \ln(x) - 1 \) becomes positive, and as \( x \) increases, \( f(x) \) decreases towards 0 from the positive side.

- **At large \( x \) (\( x \to \infty \)):** \( \ln(x) \) increases without bound, so \( \ln(x) - 1 \) becomes very large, and \( f(x) \) approaches 0 from the positive side.

### Step 3: Combine the information to determine the range
Considering the behavior of the function, the range of \( f(x) \) excludes the value 0, but it includes all other real numbers. Specifically, \( f(x) \) can take any value from \( -\infty \) to \( 0 \) (excluding 0) when \( x \) is just less than \( e \), and any value from \( 0 \) (excluding 0) to \( +\infty \) when \( x \) is just greater than \( e \).

### Conclusion:
The range of \( f(x) = \frac{1}{\ln(x) - 1} \) is \( (-\infty, 0) \cup (0, +\infty) \).

Would you like a more detailed explanation or have any questions?

Here are some related questions to consider:
1. What is the derivative of the function \( f(x) = \frac{1}{\ln(x) - 1} \)?
2. How would the graph of \( f(x) = \frac{1}{\ln(x) - 1} \) look, and what are its asymptotes?
3. What happens to the function \( f(x) \) as \( x \) approaches \( e^2 \)?
4. Can you find the inverse of \( f(x) = \frac{1}{\ln(x) - 1} \)?
5. How does the function \( g(x) = \frac{1}{\ln(x) + 1} \) compare to \( f(x) \)?

**Tip:** When analyzing the range of a function, it's often helpful to consider the behavior of the function as the input approaches critical points, such as where the function is undefined or where the function's derivative changes sign.

Explore the range of the function f(x) = 1 / (ln(x) - 1). This problem delves into logarithmic functions, analyzing their behavior and how it influences the range. Suitable for students in Grades 11-12.

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