The expression you've provided is:

\[
\log\left(x\left(x - 1\right)\right)
\]

To simplify or manipulate this expression, we can rewrite it step by step.

1. First, expand the product inside the logarithm:
   \[
   x(x - 1) = x^2 - x
   \]
   
   So, the expression becomes:
   \[
   \log\left(x^2 - x\right)
   \]

This is already a simplified form. However, keep in mind that the domain of the logarithmic function requires that the argument (inside the logarithm) must be positive:

\[
x^2 - x > 0
\]

Factoring the quadratic expression:
\[
x(x - 1) > 0
\]

From this inequality, we can conclude that \( x > 1 \) or \( x < 0 \), meaning the logarithmic function is only valid when \( x \) lies in those intervals.

Would you like further details on solving the inequality or have any other questions?

### Related Questions:
1. What are the properties of logarithmic functions and their domains?
2. How do you solve logarithmic equations involving products?
3. Can we express the sum or difference of logarithms instead of the product inside the log?
4. What happens if \( x \) is between 0 and 1 in this logarithmic expression?
5. How can we find the derivative of \( \log(x(x - 1)) \)?

**Tip:** Always check the domain restrictions when working with logarithmic functions to ensure the argument is positive.

Explore how to simplify and solve the logarithmic expression \log(x(x - 1)). Learn about logarithmic properties, quadratic factoring, and the domain restrictions of logarithmic functions. Suitable for students in Grades 9-12.

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