If `x,y,z` are in H.P then the value of expression `log(x+z)+log(x-2y+z)=`

Options

A. log (x-z)
B. 2log(x-z)
C. 3log(x-z)
D. 4log(x-z)

Answer

To solve the problem, we need to find the value of the expression \( \log(x+z) + \log(x-2y+z) \) given that \( x, y, z \) are in Harmonic Progression (H.P).

### Step-by-step Solution:

1. **Understanding H.P**:
If \( x, y, z \) are in H.P, then their reciprocals \( \frac{1}{x}, \frac{1}{y}, \frac{1}{z} \) are in Arithmetic Progression (A.P). This means:
\[ \frac{1}{y} = \frac{1}{2} \left( \frac{1}{x} + \frac{1}{z} \right) \]

2. **Finding \( y \)**:
From the A.P condition, we can derive:
\[ 2 \cdot \frac{1}{y} = \frac{1}{x} + \frac{1}{z} \]
Rearranging gives:
\[ \frac{2}{y} = \frac{z + x}{xz} \]
Thus, we can express \( y \) as:
\[ y = \frac{2xz}{x + z} \]

3. **Substituting \( y \) into the expression**:
Now we substitute \( y \) into the expression \( \log(x+z) + \log(x-2y+z) \):
\[ \log(x+z) + \log\left(x - 2\left(\frac{2xz}{x+z}\right) + z\right) \]

4. **Simplifying the second logarithm**:
The expression inside the second logarithm becomes:
\[ x - \frac{4xz}{x+z} + z \]
Finding a common denominator:
\[ = \frac{(x+z)(x) - 4xz + (x+z)(z)}{x+z} = \frac{x^2 + xz + z^2 - 4xz}{x+z} = \frac{x^2 + z^2 - 3xz}{x+z} \]

5. **Combining the logarithms**:
Now we can combine the logarithms using the property \( \log a + \log b = \log(ab) \):
\[ \log\left((x+z) \cdot \frac{x^2 + z^2 - 3xz}{x+z}\right) = \log(x^2 + z^2 - 3xz) \]

6. **Recognizing a perfect square**:
The expression \( x^2 + z^2 - 3xz \) can be rewritten as:
\[ x^2 - 2xz + z^2 = (x - z)^2 \]

7. **Final logarithmic expression**:
Thus, we have:
\[ \log((x - z)^2) = 2\log|x - z| \]

### Final Answer:
The value of the expression \( \log(x+z) + \log(x-2y+z) \) is:
\[ 2\log|x - z| \]