We can simplify the expression \( \log(5) + \log(x^2 - 1) - \log(x - 1) \) using logarithmic properties.

### Step-by-step solution:

1. **Apply the product rule of logarithms:**
   \[
   \log(a) + \log(b) = \log(a \cdot b)
   \]
   So, we combine \( \log(5) + \log(x^2 - 1) \) as:
   \[
   \log(5(x^2 - 1))
   \]

2. **Apply the quotient rule of logarithms:**
   \[
   \log(a) - \log(b) = \log\left(\frac{a}{b}\right)
   \]
   Now, we subtract \( \log(x - 1) \):
   \[
   \log\left(\frac{5(x^2 - 1)}{x - 1}\right)
   \]

3. **Simplify the expression:**
   Notice that \( x^2 - 1 \) can be factored as:
   \[
   x^2 - 1 = (x - 1)(x + 1)
   \]
   So, the expression becomes:
   \[
   \log\left(\frac{5(x - 1)(x + 1)}{x - 1}\right)
   \]

4. **Cancel common terms:**
   The \( x - 1 \) in the numerator and denominator cancels out (as long as \( x \neq 1 \)):
   \[
   \log(5(x + 1))
   \]

Thus, the simplified form is:
\[
\log(5(x + 1))
\]

### Final answer:
\[
\log(5(x + 1))
\]

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

Here are 5 related questions:
1. How does the quotient rule of logarithms work?
2. What are other logarithmic identities useful for simplification?
3. How do you solve logarithmic equations involving sums and differences?
4. How can you determine the domain of a logarithmic expression like \( \log(5(x + 1)) \)?
5. What happens if \( x = 1 \) in the original expression?

**Tip:** When working with logarithms, always check for restrictions on the domain (i.e., where the argument is positive).

Discover how to simplify the logarithmic expression log(5) + log(x^2 - 1) - log(x - 1) using properties of logarithms and factorization. Learn step-by-step techniques with the final solution log(5(x + 1)), suitable for high school students.

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