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The image contains a multiple-choice question related to logarithms. Here's the text from the image:

**The LCM and HCF of two numbers `x` and `y` are `k` and `l` respectively. The value of `log x + log y` is:**

(A) \( \log (\text{LCM of } x \text{ and } y) + \log (\text{HCF of } x \text{ and } y) \)

(B) \( \log (\text{LCM of } x \text{ and } y) \times \log (\text{HCF of } x \text{ and } y) \)

(C) \( \log \left(\frac{\text{LCM of } x \text{ and } y}{\text{HCF of } x \text{ and } y}\right) \)

(D) \( \log \left(\text{HCF of } x \text{ and } y \times \frac{\text{LCM of } x \text{ and } y}{\text{HCF of } x \text{ and } y}\right) \)

### Explanation:
Given that the LCM and HCF of two numbers \( x \) and \( y \) are \( k \) and \( l \) respectively, we know from number theory that:
\[ x \times y = \text{LCM}(x, y) \times \text{HCF}(x, y) \]

Therefore, 
\[ x \times y = k \times l \]
Taking logarithms on both sides:
\[ \log(x \times y) = \log(k \times l) \]

Using the property of logarithms, \(\log(ab) = \log(a) + \log(b)\), we get:
\[ \log x + \log y = \log k + \log l \]

So, the correct answer corresponds to **Option A**:
\[ \log(\text{LCM of } x \text{ and } y) + \log(\text{HCF of } x \text{ and } y) \]

Would you like more details on the properties of logarithms or how the LCM and HCF relate to the product of numbers?

### Related Questions:
1. What is the formula for the relationship between LCM, HCF, and the product of two numbers?
2. How does the logarithm function behave when multiplying or dividing two numbers?
3. Can you explain how logarithms simplify multiplication into addition?
4. What is the significance of logarithms in solving algebraic equations?
5. How can you find the LCM and HCF of two numbers using prime factorization?

### Tip:
When dealing with logarithms, remember the key properties: the logarithm of a product is the sum of the logarithms, and the logarithm of a quotient is the difference of the logarithms. These properties simplify complex expressions significantly.

Understand and solve a logarithmic problem involving the LCM and HCF of two numbers. Learn about logarithm properties and their application in determining the sum of logarithms. Suitable for Grades 10-12 students.

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