If a,b,c are in GP then `log_(10)a, log_(10)b, log_(10)c` are in

Options

A. GP
B. HP
C. AP
D. None of these

Answer

To solve the problem, we need to determine the relationship between \( \log_{10} a \), \( \log_{10} b \), and \( \log_{10} c \) given that \( a, b, c \) are in geometric progression (GP).

### Step-by-Step Solution:

1. **Understanding the Condition for GP**:
Since \( a, b, c \) are in GP, the condition that must hold is:
\[ b^2 = ac \]

2. **Taking Logarithms**:
We take logarithm base 10 on both sides of the equation:
\[ \log_{10}(b^2) = \log_{10}(ac) \]

3. **Applying Logarithm Properties**:
Using the property of logarithms that states \( \log(a^b) = b \log(a) \), we can rewrite the left side:
\[ 2 \log_{10}(b) = \log_{10}(a) + \log_{10}(c) \]

4. **Rearranging the Equation**:
Rearranging the equation gives us:
\[ 2 \log_{10}(b) - \log_{10}(a) - \log_{10}(c) = 0 \]

5. **Identifying Terms**:
Let:
- \( x_1 = \log_{10}(a) \)
- \( x_2 = \log_{10}(b) \)
- \( x_3 = \log_{10}(c) \)

Then the equation can be expressed as:
\[ 2x_2 - x_1 - x_3 = 0 \]

6. **Rearranging to Identify AP**:
Rearranging gives:
\[ x_2 - x_1 = x_3 - x_2 \]
This shows that the difference between the first and second terms is equal to the difference between the second and third terms.

7. **Conclusion**:
The condition \( x_2 - x_1 = x_3 - x_2 \) indicates that \( x_1, x_2, x_3 \) are in arithmetic progression (AP).

### Final Answer:
Thus, if \( a, b, c \) are in GP, then \( \log_{10} a, \log_{10} b, \log_{10} c \) are in **AP**.

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