The numbers ` log_(180) 12, log_(2160) 12, log_(25920) 12 ` are in

Options

A. A.P.
B. G.P.
C. H.P.
D. None of the above progressions

Answer

To determine whether the numbers \( \log_{180} 12, \log_{2160} 12, \log_{25920} 12 \) are in Arithmetic Progression (AP), Geometric Progression (GP), Harmonic Progression (HP), or none of the above, we will follow these steps:

### Step 1: Rewrite the logarithms using the change of base formula
Using the property of logarithms that states \( \log_b a = \frac{\log a}{\log b} \), we can rewrite the logarithms as follows:

1. \( \log_{180} 12 = \frac{\log 12}{\log 180} \)
2. \( \log_{2160} 12 = \frac{\log 12}{\log 2160} \)
3. \( \log_{25920} 12 = \frac{\log 12}{\log 25920} \)

### Step 2: Simplify the logarithms
Next, we need to express \( \log 2160 \) and \( \log 25920 \) in terms of \( \log 180 \) and \( \log 12 \):

- \( 2160 = 180 \times 12 \)
- \( 25920 = 180 \times 12^2 \)

Thus, we can compute:

1. \( \log 2160 = \log(180 \times 12) = \log 180 + \log 12 \)
2. \( \log 25920 = \log(180 \times 12^2) = \log 180 + 2\log 12 \)

Now we can rewrite the logarithms:

1. \( \log_{180} 12 = \frac{\log 12}{\log 180} \)
2. \( \log_{2160} 12 = \frac{\log 12}{\log 180 + \log 12} \)
3. \( \log_{25920} 12 = \frac{\log 12}{\log 180 + 2\log 12} \)

### Step 3: Check for Arithmetic Progression (AP)
For the numbers to be in AP, the condition \( 2b = a + c \) must hold, where \( a = \log_{180} 12 \), \( b = \log_{2160} 12 \), and \( c = \log_{25920} 12 \).

Calculating \( b - a \) and \( c - b \):

- \( b - a = \frac{\log 12}{\log 180 + \log 12} - \frac{\log 12}{\log 180} \)
- \( c - b = \frac{\log 12}{\log 180 + 2\log 12} - \frac{\log 12}{\log 180 + \log 12} \)

After simplification, we find that these differences do not yield a constant value, thus they are not in AP.

### Step 4: Check for Geometric Progression (GP)
For the numbers to be in GP, the condition \( \frac{b}{a} = \frac{c}{b} \) must hold.

Calculating \( \frac{b}{a} \) and \( \frac{c}{b} \):

- \( \frac{b}{a} = \frac{\frac{\log 12}{\log 180 + \log 12}}{\frac{\log 12}{\log 180}} = \frac{\log 180}{\log 180 + \log 12} \)
- \( \frac{c}{b} = \frac{\frac{\log 12}{\log 180 + 2\log 12}}{\frac{\log 12}{\log 180 + \log 12}} = \frac{\log 180 + \log 12}{\log 180 + 2\log 12} \)

After simplification, we find that these ratios are not equal, thus they are not in GP.

### Step 5: Check for Harmonic Progression (HP)
For the numbers to be in HP, the reciprocals must be in AP. The reciprocals are:

1. \( \frac{\log 180}{\log 12} \)
2. \( \frac{\log 180 + \log 12}{\log 12} \)
3. \( \frac{\log 180 + 2\log 12}{\log 12} \)

We check if:

\[ 2 \cdot \frac{\log 180 + \log 12}{\log 12} = \frac{\log 180}{\log 12} + \frac{\log 180 + 2\log 12}{\log 12} \]

After simplification, we find that this condition holds true, thus the numbers are in HP.

### Conclusion
The numbers \( \log_{180} 12, \log_{2160} 12, \log_{25920} 12 \) are in Harmonic Progression (HP).