`log_(3) 2, log_(6) 2, log_(12)2` are in

Options

A. A.P.
B. G.P.
C. H.P.
D. none

Answer

To determine whether the logarithmic values \( \log_{3} 2, \log_{6} 2, \log_{12} 2 \) are in Harmonic Progression (HP), we can follow these steps:

### Step 1: Convert the logarithms to a common base
Using the change of base formula, we can express each logarithm in terms of base 2:
\[ \log_{3} 2 = \frac{1}{\log_{2} 3}, \quad \log_{6} 2 = \frac{1}{\log_{2} 6}, \quad \log_{12} 2 = \frac{1}{\log_{2} 12} \]

### Step 2: Simplify the logarithms
Next, we can express \( \log_{2} 6 \) and \( \log_{2} 12 \) using properties of logarithms:
\[ \log_{2} 6 = \log_{2} (2 \cdot 3) = \log_{2} 2 + \log_{2} 3 = 1 + \log_{2} 3 \]
\[ \log_{2} 12 = \log_{2} (4 \cdot 3) = \log_{2} 4 + \log_{2} 3 = 2 + \log_{2} 3 \]

### Step 3: Substitute back into the logarithmic expressions
Now we substitute these back into our expressions:
\[ \log_{3} 2 = \frac{1}{\log_{2} 3}, \quad \log_{6} 2 = \frac{1}{1 + \log_{2} 3}, \quad \log_{12} 2 = \frac{1}{2 + \log_{2} 3} \]

### Step 4: Let \( x = \log_{2} 3 \)
Let \( x = \log_{2} 3 \). Then we can rewrite our logarithmic values as:
\[ \log_{3} 2 = \frac{1}{x}, \quad \log_{6} 2 = \frac{1}{1 + x}, \quad \log_{12} 2 = \frac{1}{2 + x} \]

### Step 5: Check for Harmonic Progression
For three numbers \( a, b, c \) to be in HP, the reciprocals \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) must be in Arithmetic Progression (AP). Therefore, we need to check if:
\[ \frac{1}{\log_{3} 2}, \frac{1}{\log_{6} 2}, \frac{1}{\log_{12} 2} \]
are in AP, which means:
\[ 2 \cdot \frac{1}{1 + x} = \frac{1}{x} + \frac{1}{2 + x} \]

### Step 6: Simplify the equation
Cross-multiplying gives:
\[ 2(2 + x) = (1 + x)(2 + x) \]
Expanding both sides:
\[ 4 + 2x = 2 + 3x + x^2 \]
Rearranging gives:
\[ x^2 + x - 2 = 0 \]

### Step 7: Factor the quadratic equation
Factoring the quadratic:
\[ (x - 1)(x + 2) = 0 \]
Thus, \( x = 1 \) or \( x = -2 \).

### Step 8: Conclusion
Since we found that the condition for HP holds, we conclude that \( \log_{3} 2, \log_{6} 2, \log_{12} 2 \) are in Harmonic Progression.