Question

In the formula \(log_a xy=log_a x+log_a y\), which of the following is true?

• A. x>0, y>0, a=1
• B. x<0, y<0, a=1
• C. a>0, y>0, x=1
• D. x>0, y>0, a ≠1

Answer 1

The Correct Option is D

To solve this question, we need to analyze the given logarithmic identity:

1. Given Formula:
The logarithmic identity is:
\[ \log_a(xy) = \log_a(x) + \log_a(y) \]

2. Conditions for the Identity to Hold:
For the identity to be valid, the following conditions must hold true: - \( x > 0 \), \( y > 0 \) (since the logarithm of a non-positive number is not defined). - \( a > 0 \) and \( a \neq 1 \), as the base of a logarithm must be positive and not equal to 1.

3. Identifying the Correct Option:
- Option (A) is incorrect because the value of \( a \) must not be equal to 1.
- Option (B) is incorrect because \( x \) and \( y \) should be positive.
- Option (C) is incomplete because the values of \( x \) and \( y \) are not defined as positive.
- Option (D) is correct as it specifies that \( x > 0 \), \( y > 0 \), and \( a \neq 1 \), which is the required condition for the identity to hold.

Final Answer:
Option (D) \( x > 0, y > 0, a \neq 1 \) is correct.