If `x^(18)=y^(21)=z^(28)`, then 3,3 `log_(y)x,3log_(z)y,7log_(x)z` are in

Options

A. A.P.
B. G.P.
C. H.P.
D. A.G.P.

Answer

To solve the problem where \( x^{18} = y^{21} = z^{28} \), we will find out if \( 3 \log_y x \), \( 3 \log_z y \), and \( 7 \log_x z \) are in arithmetic progression (AP).

### Step-by-step Solution:

1. **Set a Common Value**:
Let \( x^{18} = y^{21} = z^{28} = k \).

2. **Take Logarithms**:
Taking logarithms on both sides, we have:
\[ 18 \log x = 21 \log y = 28 \log z = \log k \]

3. **Express Logarithms**:
From the above equations, we can express the logarithms as:
\[ \log x = \frac{\log k}{18}, \quad \log y = \frac{\log k}{21}, \quad \log z = \frac{\log k}{28} \]

4. **Find Ratios**:
Now, we can find the ratios:
- For \( \log_y x \):
\[ \log_y x = \frac{\log x}{\log y} = \frac{\frac{\log k}{18}}{\frac{\log k}{21}} = \frac{21}{18} = \frac{7}{6} \]
- For \( \log_z y \):
\[ \log_z y = \frac{\log y}{\log z} = \frac{\frac{\log k}{21}}{\frac{\log k}{28}} = \frac{28}{21} = \frac{4}{3} \]
- For \( \log_x z \):
\[ \log_x z = \frac{\log z}{\log x} = \frac{\frac{\log k}{28}}{\frac{\log k}{18}} = \frac{18}{28} = \frac{9}{14} \]

5. **Calculate the Values**:
Now we can calculate:
- \( 3 \log_y x = 3 \cdot \frac{7}{6} = \frac{21}{6} = \frac{7}{2} \)
- \( 3 \log_z y = 3 \cdot \frac{4}{3} = 4 \)
- \( 7 \log_x z = 7 \cdot \frac{9}{14} = \frac{63}{14} = \frac{9}{2} \)

6. **Check for Arithmetic Progression**:
Now we have the three terms:
- \( a = \frac{7}{2} \)
- \( b = 4 \)
- \( c = \frac{9}{2} \)

To check if they are in AP, we need to see if:
\[ 2b = a + c \]
Substituting the values:
\[ 2 \cdot 4 = \frac{7}{2} + \frac{9}{2} \]
Simplifying the right-hand side:
\[ 8 = \frac{16}{2} = 8 \]
Since both sides are equal, the terms are in AP.

### Conclusion:
Thus, \( 3 \log_y x \), \( 3 \log_z y \), and \( 7 \log_x z \) are in arithmetic progression.