If log a, log b, and log c are in A.P. and also log a - log 2b, log 2b - log 3c, log 3c - log a are in A.P., then

Options

A. a, b, c, are in H.P.
B. a, 2b, 3c are in A.P.
C. a, b, c are the sides of a triangle
D. none of the above

Answer

To solve the problem, we need to analyze the conditions given in the question step by step.

### Step 1: Understand the first condition
We are given that \( \log a, \log b, \log c \) are in Arithmetic Progression (A.P.).

In an A.P., the middle term is the average of the other two terms. Therefore, we can write:
\[ 2 \log b = \log a + \log c \]

### Step 2: Use properties of logarithms
Using the properties of logarithms, we can rewrite the equation:
\[ \log b^2 = \log(ac) \]
This implies:
\[ b^2 = ac \]
This shows that \( a, b, c \) are in Geometric Progression (G.P.).

### Step 3: Understand the second condition
Next, we have the second condition where \( \log a - \log 2b, \log 2b - \log 3c, \log 3c - \log a \) are also in A.P.

Let’s denote:
\[ x_1 = \log a - \log 2b = \log \left(\frac{a}{2b}\right) \]
\[ x_2 = \log 2b - \log 3c = \log \left(\frac{2b}{3c}\right) \]
\[ x_3 = \log 3c - \log a = \log \left(\frac{3c}{a}\right) \]

Since these terms are in A.P., we have:
\[ 2x_2 = x_1 + x_3 \]

### Step 4: Substitute the expressions
Substituting the expressions we found:
\[ 2 \log \left(\frac{2b}{3c}\right) = \log \left(\frac{a}{2b}\right) + \log \left(\frac{3c}{a}\right) \]

Using properties of logarithms, we can combine the right-hand side:
\[ 2 \log \left(\frac{2b}{3c}\right) = \log \left(\frac{a \cdot 3c}{2b \cdot a}\right) = \log \left(\frac{3c}{2b}\right) \]

### Step 5: Simplifying the equation
This leads to:
\[ \log \left(\frac{2b}{3c}\right)^2 = \log \left(\frac{3c}{2b}\right) \]
Taking the antilogarithm gives:
\[ \left(\frac{2b}{3c}\right)^2 = \frac{3c}{2b} \]

### Step 6: Cross-multiplying and simplifying
Cross-multiplying gives:
\[ 4b^2 = 9c^2 \]
This implies:
\[ \frac{b^2}{c^2} = \frac{9}{4} \quad \Rightarrow \quad \frac{b}{c} = \frac{3}{2} \]

### Step 7: Relate \( a, b, c \)
From \( b^2 = ac \) and \( \frac{b}{c} = \frac{3}{2} \), we can express \( b \) in terms of \( c \):
\[ b = \frac{3}{2}c \]
Substituting into \( b^2 = ac \):
\[ \left(\frac{3}{2}c\right)^2 = ac \quad \Rightarrow \quad \frac{9}{4}c^2 = ac \quad \Rightarrow \quad a = \frac{9}{4}c \]

### Step 8: Finding the ratio \( a : b : c \)
Now we have:
- \( a = \frac{9}{4}c \)
- \( b = \frac{3}{2}c \)
- \( c = c \)

To find the ratio \( a : b : c \):
\[ a : b : c = \frac{9}{4}c : \frac{3}{2}c : c \]
Dividing by \( c \):
\[ = \frac{9}{4} : \frac{3}{2} : 1 \]
To simplify, multiply through by 4:
\[ = 9 : 6 : 4 \]

### Conclusion
Thus, the final ratio \( a : b : c \) is \( 9 : 6 : 4 \).