If `log_(y)x=(a.log_(z)y)=(b.log_(x)z)=ab`, then which of the following pairs of values for (a,b) is not possible?

Options

A. `(-2,1//2)`
B. `(1,1)`
C. `(0.4,2.5)`
D. `(2,2)`

Answer

To solve the problem, we need to analyze the given logarithmic equations and derive the possible values for \( ab \).

### Step-by-Step Solution:

1. **Given Equations**:
We have the following equations based on the problem statement:
\[ \log_y x = ab \]
\[ a \cdot \log_z y = ab \]
\[ b \cdot \log_x z = ab \]

2. **Rearranging the Second Equation**:
From the second equation, we can express \( \log_z y \) in terms of \( a \):
\[ \log_z y = \frac{ab}{a} = b \]

3. **Rearranging the Third Equation**:
From the third equation, we can express \( \log_x z \) in terms of \( b \):
\[ \log_x z = \frac{ab}{b} = a \]

4. **Using Change of Base Formula**:
We can use the change of base formula to express the logarithms in terms of each other:
\[ \log_y x = \frac{1}{\log_x y} \]
\[ \log_z y = \frac{1}{\log_y z} \]
\[ \log_x z = \frac{1}{\log_z x} \]

5. **Substituting Values**:
Substitute the expressions we found:
\[ \log_y x = ab \]
\[ \log_z y = b \]
\[ \log_x z = a \]

6. **Combining the Logarithmic Relationships**:
From the relationships, we can write:
\[ \log_y x \cdot \log_z y \cdot \log_x z = 1 \]
Substituting the values:
\[ ab \cdot b \cdot a = 1 \]
This simplifies to:
\[ a^2 b^2 = 1 \]

7. **Finding the Values of \( ab \)**:
Taking the square root of both sides gives:
\[ ab = \pm 1 \]

8. **Checking the Options**:
Now we need to check the given options for pairs \( (a, b) \) to see which one does not yield \( ab = \pm 1 \):
- **Option 1**: \( a = -2, b = \frac{1}{2} \) → \( ab = -1 \) (Possible)
- **Option 2**: \( a = 1, b = 1 \) → \( ab = 1 \) (Possible)
- **Option 3**: \( a = 0.4, b = 2.5 \) → \( ab = 1 \) (Possible)
- **Option 4**: \( a = 2, b = 2 \) → \( ab = 4 \) (Not Possible)

### Conclusion:
The pair of values for \( (a, b) \) that is not possible is **Option 4: \( (2, 2) \)**.