Consider this function, defined for every x >1 (where "In" indicates the 
natural logarithm): f(x) =2ln (x) -2ln (x-1). Which of these is an expression for 
its inverse function 𝑓−1(𝑦)

We are tasked with finding the inverse of the function \( f(x) = 2 \ln(x) - 2 \ln(x-1) \). Here's a step-by-step breakdown to find the inverse function \( f^{-1}(y) \):

### Step 1: Set the equation
We begin by rewriting the function and setting \( y = f(x) \):
\[
y = 2 \ln(x) - 2 \ln(x-1)
\]

### Step 2: Simplify the right-hand side
Using logarithmic properties, we can simplify the right-hand side:
\[
y = 2 \ln \left( \frac{x}{x-1} \right)
\]
We can further simplify by dividing both sides by 2:
\[
\frac{y}{2} = \ln \left( \frac{x}{x-1} \right)
\]

### Step 3: Exponentiate both sides
Exponentiate both sides to eliminate the natural logarithm:
\[
e^{\frac{y}{2}} = \frac{x}{x-1}
\]

### Step 4: Solve for \( x \)
Now solve for \( x \). Multiply both sides by \( x - 1 \) to clear the denominator:
\[
e^{\frac{y}{2}} (x - 1) = x
\]
Expand:
\[
e^{\frac{y}{2}} x - e^{\frac{y}{2}} = x
\]
Rearrange to collect the terms with \( x \):
\[
e^{\frac{y}{2}} x - x = e^{\frac{y}{2}}
\]
Factor out \( x \) on the left-hand side:
\[
x (e^{\frac{y}{2}} - 1) = e^{\frac{y}{2}}
\]
Finally, solve for \( x \):
\[
x = \frac{e^{\frac{y}{2}}}{e^{\frac{y}{2}} - 1}
\]

### Final Answer:
Thus, the inverse function \( f^{-1}(y) \) is:
\[
f^{-1}(y) = \frac{e^{\frac{y}{2}}}{e^{\frac{y}{2}} - 1}
\]

---

Would you like further details or explanations?

Here are 5 related questions for further exploration:
1. How do logarithmic properties help in simplifying functions?
2. Why do we use exponentiation to eliminate natural logarithms?
3. What is the domain of the inverse function \( f^{-1}(y) \)?
4. How can we verify if \( f^{-1}(y) \) is correct by composition \( f(f^{-1}(y)) \)?
5. How would the process change if the function had an additional constant term?

**Tip:** Always check the domain of both the original and inverse functions when solving problems involving inverses.

Consider this function, defined for every x >1 (where "In" indicates the natural logarithm): f(x) = 2ln(x) - 2ln(x-1). Which of these is an expression for its inverse function f⁻¹(y)?

Learn how to find the inverse of the logarithmic function f(x) = 2ln(x) - 2ln(x-1). This step-by-step solution simplifies logarithmic expressions, applies exponential functions, and solves for the inverse function f⁻¹(y). Suitable for advanced high school and college-level math students.

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