log(x²y²) – 2log(x) = ?

The correct answer is 2log y. To simplify the expression log(x²y²) - 2log(x), we need to apply the fundamental properties of logarithms. The relevant properties are:

• The product rule: log(AB) = log A + log B
• The power rule: log(Aⁿ) = n log A

Let's apply these rules step-by-step:

1. First, apply the product rule to the term log(x²y²):
   log(x²y²) = log(x²) + log(y²)
2. Next, apply the power rule to both log(x²) and log(y²), and also to the term 2log(x):
   log(x²) becomes 2log(x)
   log(y²) becomes 2log(y)
   The expression 2log(x) remains as is.
3. Substitute these back into the original expression:
   (2log(x) + 2log(y)) - 2log(x)
4. Now, simplify by combining like terms:
   2log(x) + 2log(y) - 2log(x) = 2log(y)

Therefore, the simplified expression is 2log y.

• Option A, y(x²y² - 2x), is incorrect. This expression does not follow any standard logarithm rules and is an algebraic manipulation that doesn't apply here.
• Option C, log(xy³), is incorrect. This would imply operations like log(y²) + log(y) which is not what the given expression simplifies to.
• Option D, 2log(x²y³), is incorrect. This result would require different initial terms or operations and does not match the correct simplification process.