To solve the problem where \(1, \log_y x, \log_z y, -15 \log_x z\) are in Arithmetic Progression (A.P.), we will follow these steps:
### Step 1: Understanding the A.P. Condition
In an A.P., the middle term is the average of the other two terms. Therefore, we can express the condition as:
\[
\log_y x = \frac{1 + (-15 \log_x z)}{2}
\]
### Step 2: Expressing Logarithms in Terms of Base 10
Using the change of base formula, we can express the logarithms in terms of a common base (let's use base 10):
\[
\log_y x = \frac{\log x}{\log y}, \quad \log_z y = \frac{\log y}{\log z}, \quad \log_x z = \frac{\log z}{\log x}
\]
### Step 3: Substitute the Logarithmic Expressions
Substituting these into the A.P. condition gives:
\[
\frac{\log x}{\log y} = \frac{1 - 15 \frac{\log z}{\log x}}{2}
\]
### Step 4: Cross-Multiplying to Eliminate Fractions
Cross-multiplying yields:
\[
2 \log x \cdot \log x = (1 - 15 \frac{\log z}{\log x}) \log y
\]
\[
2 (\log x)^2 = \log y - 15 \log z
\]
### Step 5: Rearranging the Equation
Rearranging the equation gives:
\[
2 (\log x)^2 + 15 \log z = \log y
\]
### Step 6: Expressing \(y\) in Terms of \(x\) and \(z\)
From the equation, we can express \(y\) as:
\[
y = 2 (\log x)^2 + 15 \log z
\]
### Step 7: Finding Relationships
Now we can also express \(z\) in terms of \(x\) and \(y\) using the earlier logarithmic expressions:
\[
\log_z y = \frac{\log y}{\log z}
\]
Substituting \(y\) into this gives us a relationship between \(x\), \(y\), and \(z\).
### Step 8: Solving for \(x\), \(y\), and \(z\)
We can set up equations based on the relationships derived and solve for \(x\), \(y\), and \(z\).
### Conclusion
The relationships derived from the A.P. condition will lead us to conclude that \(x\), \(y\), and \(z\) are related in a specific way, which can be further simplified to find the exact values or ratios.
