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.