If `xgey` and `ygt1`, then the value of the expression `log_(x)(x/y)+log_(y)(y/x)` can never be

Options

A. `-1`
B. `-0.5`
C. 0
D. 1

Answer

To solve the expression \( \log_{x}\left(\frac{x}{y}\right) + \log_{y}\left(\frac{y}{x}\right) \) under the conditions \( x \geq y \) and \( y > 1 \), we can follow these steps:

### Step 1: Rewrite the Logarithmic Expressions
Using the property of logarithms that states \( \log_{a}\left(\frac{b}{c}\right) = \log_{a}(b) - \log_{a}(c) \), we can rewrite the expression:

\[ \log_{x}\left(\frac{x}{y}\right) = \log_{x}(x) - \log_{x}(y) \]
\[ \log_{y}\left(\frac{y}{x}\right) = \log_{y}(y) - \log_{y}(x) \]

### Step 2: Substitute the Values
Substituting these back into the expression gives:

\[ \log_{x}(x) - \log_{x}(y) + \log_{y}(y) - \log_{y}(x) \]

Since \( \log_{x}(x) = 1 \) and \( \log_{y}(y) = 1 \), we can simplify this to:

\[ 1 - \log_{x}(y) + 1 - \log_{y}(x) \]

This simplifies further to:

\[ 2 - \log_{x}(y) - \log_{y}(x) \]

### Step 3: Use the Change of Base Formula
Using the change of base formula, we can express \( \log_{y}(x) \) in terms of \( \log_{x}(y) \):

\[ \log_{y}(x) = \frac{1}{\log_{x}(y)} \]

Thus, we can rewrite our expression as:

\[ 2 - \log_{x}(y) - \frac{1}{\log_{x}(y)} \]

### Step 4: Let \( t = \log_{x}(y) \)
Letting \( t = \log_{x}(y) \), we can rewrite the expression as:

\[ 2 - t - \frac{1}{t} \]

### Step 5: Find the Minimum Value
To find the minimum value of the expression \( 2 - t - \frac{1}{t} \), we can use the AM-GM inequality:

\[ t + \frac{1}{t} \geq 2 \]

Thus,

\[ 2 - (t + \frac{1}{t}) \leq 2 - 2 = 0 \]

This means the minimum value of \( 2 - t - \frac{1}{t} \) is \( 0 \).

### Conclusion
The expression \( \log_{x}\left(\frac{x}{y}\right) + \log_{y}\left(\frac{y}{x}\right) \) can never be less than \( 0 \). Therefore, the value of the expression can never be negative.