If x ≥ y and y > 1, then the value of the expression logx + logy  can never be

-1

-0.5

0

1

Given the constraints \( x \geq y \) and \( y > 1 \), we need to analyze the values of the expression \( \log x + \log y \).

First, let's recall the logarithmic property:
\[
\log a + \log b = \log(ab)
\]
So, the expression \( \log x + \log y \) can be rewritten as:
\[
\log(xy)
\]

Given \( y > 1 \), we know \( y \) is a positive number greater than 1. Also, since \( x \geq y \), \( x \) is at least 1 and could be equal to or greater than \( y \).

Let's evaluate the possible values of \( \log(xy) \):

1. **Can \(\log(xy) = -1\)?**

    For \(\log(xy) = -1\), we need:
    \[
    xy = 10^{-1} = 0.1
    \]
    Since \( y > 1 \), \( y \) cannot be less than or equal to 1, making \( xy \geq y \), which is always greater than 1. Thus, \( xy = 0.1 \) is not possible under the given constraints.

2. **Can \(\log(xy) = -0.5\)?**

    For \(\log(xy) = -0.5\), we need:
    \[
    xy = 10^{-0.5} \approx 0.316
    \]
    Similarly, since \( y > 1 \), \( y \) cannot be less than or equal to 1, making \( xy \geq y \), which is always greater than 1. Thus, \( xy = 0.316 \) is not possible under the given constraints.

3. **Can \(\log(xy) = 0\)?**

    For \(\log(xy) = 0\), we need:
    \[
    xy = 10^0 = 1
    \]
    If \( y > 1 \), then \( x \geq y \) implies \( x \) and \( y \) are both greater than or equal to 1. Therefore, the product \( xy \geq 1 \times 1 = 1 \). Thus, \( xy = 1 \) is possible when \( x = y = 1 \), but since \( y > 1 \), this case cannot happen. Therefore, \( xy = 1 \) is not possible.

4. **Can \(\log(xy) = 1\)?**

    For \(\log(xy) = 1\), we need:
    \[
    xy = 10^1 = 10
    \]
    Since \( y > 1 \) and \( x \geq y \), \( x \) could be such that \( xy = 10 \). For example, if \( y = 2 \), \( x \) could be 5, making \( xy = 10 \). Thus, \( xy = 10 \) is possible.

Given this analysis, the value that the expression \( \log x + \log y \) can never be is:

\[
\boxed{-1}
\]

Would you like further details on this explanation, or do you have any questions? Here are some related questions for practice:

1. If \( x \geq 2 \) and \( y = 2 \), what are the possible values of \( \log x + \log y \)?
2. Can \( \log x + \log y \) be a negative value if \( x \geq y > 1 \)?
3. If \( x = y \) and \( y > 1 \), what is the minimum value of \( \log x + \log y \)?
4. What is the value of \( \log x + \log y \) if \( x = 2 \) and \( y = 3 \)?
5. If \( \log x + \log y = 1 \) and \( y > 1 \), what are the possible values of \( x \) and \( y \)?
6. How does the value of \( \log x + \log y \) change if \( x \) and \( y \) are both doubled?
7. Can the expression \( \log x + \log y \) equal zero if \( x \geq y \) and \( y > 1 \)?
8. What are the implications for \( x \) and \( y \) if \( \log x + \log y = 2 \)?

**Tip:** When dealing with logarithmic expressions, converting them into simpler forms using logarithmic properties can help in solving them efficiently.

Explore why the expression \( \log x + \log y \) can never be -1, along with the analysis for -0.5, 0, and 1. Understand the constraints \( x \geq y \) and \( y > 1 \) and how they affect the possible values of \( \log x + \log y \). Suitable for high school students studying logarithmic functions and inequalities.

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