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We are tasked with finding the domain of the given function:

\[
f(x) = \ln(x^2 - 2x + 1) \cdot \ln(x - 1) \cdot \frac{x + 3}{\sqrt{x + 4}}
\]

### Step 1: Domain Restrictions for Logarithmic Functions
Logarithmic functions are defined only for positive arguments. Therefore, the arguments inside each logarithmic function must be strictly greater than zero.

#### First Logarithm: \(\ln(x^2 - 2x + 1)\)
The argument here is \(x^2 - 2x + 1\), which is a perfect square:
\[
x^2 - 2x + 1 = (x - 1)^2
\]
This is always non-negative, and it is equal to 0 when \(x = 1\). Since the logarithm requires the argument to be positive, the condition here is:
\[
(x - 1)^2 > 0 \quad \text{or} \quad x \neq 1
\]

#### Second Logarithm: \(\ln(x - 1)\)
For this term, we need:
\[
x - 1 > 0 \quad \Rightarrow \quad x > 1
\]

### Step 2: Domain Restriction for the Square Root
The square root in the denominator requires its argument to be non-negative, and it cannot be zero because it's in the denominator. Therefore, we need:
\[
x + 4 > 0 \quad \Rightarrow \quad x > -4
\]

### Step 3: Combined Domain
We must satisfy all of the following conditions:
1. \(x > 1\) from \(\ln(x - 1)\),
2. \(x \neq 1\) from \((x - 1)^2 > 0\),
3. \(x > -4\) from \(\sqrt{x + 4}\).

The strictest condition is \(x > 1\), so the domain of the function is:
\[
\boxed{(1, \infty)}
\]

Would you like further details on any of these steps?

Here are 5 related questions:
1. How would the domain change if the square root were not in the denominator?
2. Can logarithmic functions ever take negative values as inputs?
3. What happens to the domain if the square root argument is inside a logarithm?
4. How do different operations (multiplication, division) affect domain restrictions?
5. Can you provide an example where the domain includes a union of intervals?

**Tip**: Always pay attention to both logarithmic and square root terms when finding domains, as they impose strict conditions on the function's inputs.

Find the domain of f(x) = ln(x^2 - 2x + 1) * ln(x - 1) * (x + 3)/sqrt(x + 4)

Find the domain of the function f(x) = ln(x^2 - 2x + 1) * ln(x - 1) * (x + 3)/sqrt(x + 4). Learn how logarithmic and square root functions impose domain restrictions on this complex function. Suitable for undergraduate students studying calculus or advanced algebra.

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