If 1, `log_(y)x, log_(z)y,-15 log_(x)z` are in A.P. then the correct statement is :

Options

A. `z^(3)=x`
B. `x^(3)=z`
C. `z^(3)=y`
D. None of these

Explanation

To solve the problem where the numbers \( 1, \log_y x, \log_z y, -15 \log_x z \) are in Arithmetic Progression (A.P.), we will follow these steps:

### Step 1: Understand the condition for A.P.
For four numbers \( a, b, c, d \) to be in A.P., the condition is:
\[ 2b = a + c \quad \text{and} \quad 2c = b + d \]
In our case, let:
- \( a = 1 \)
- \( b = \log_y x \)
- \( c = \log_z y \)
- \( d = -15 \log_x z \)

### Step 2: Set up the first condition
Using the first condition \( 2b = a + c \):
\[ 2 \log_y x = 1 + \log_z y \]

### Step 3: Rewrite logarithms using change of base formula
Using the change of base formula, we can express the logarithms in terms of natural logarithms:
\[ \log_y x = \frac{\log x}{\log y}, \quad \log_z y = \frac{\log y}{\log z} \]
Substituting these into the equation gives:
\[ 2 \cdot \frac{\log x}{\log y} = 1 + \frac{\log y}{\log z} \]

### Step 4: Multiply through by \(\log y \cdot \log z\) to eliminate denominators
\[ 2 \log x \cdot \log z = \log y + \log^2 y \]

### Step 5: Set up the second condition
Now, using the second condition \( 2c = b + d \):
\[ 2 \log_z y = \log_y x - 15 \log_x z \]
Substituting the logarithmic expressions:
\[ 2 \cdot \frac{\log y}{\log z} = \frac{\log x}{\log y} - 15 \cdot \frac{\log z}{\log x} \]

### Step 6: Multiply through by \(\log z \cdot \log x\) to eliminate denominators
\[ 2 \log y \cdot \log x = \log x \cdot \log x - 15 \log z \cdot \log z \]

### Step 7: Rearranging the equations
We now have two equations:
1. \( 2 \log x \cdot \log z = \log y + \log^2 y \)
2. \( 2 \log y \cdot \log x = \log^2 x - 15 \log^2 z \)

### Step 8: Solve the system of equations
From the first equation, express \( \log y \) in terms of \( \log x \) and \( \log z \):
\[ \log^2 y = 2 \log x \cdot \log z - \log y \]
This is a quadratic in \( \log y \).

### Step 9: Substitute back to find relationships
Substituting \( \log y \) back into the second equation will give us a relationship between \( \log x, \log y, \) and \( \log z \).

### Conclusion
After solving the equations, we can find the relationships among \( x, y, z \). The final result will show that \( x, y, z \) are related in a specific way, confirming the statement that needs to be validated.