Question

If \( a, b, c \) elements are in geometric progression, then \( \log a, \log b, \log c \) will be in:

• A. arithmetic progression
• B. geometric progression
• C. harmonic progression
• D. arithmetic geometric progression

Hint:

For geometric progressions, the logarithms of the terms form an arithmetic progression. Always check the relation between the terms before concluding.

Answer 1

The Correct Option is A

Step 1: Condition of Geometric Progression
Since \( a, b, c \) are in geometric progression, by the definition of geometric progression, we know that the ratio between consecutive terms is constant. Therefore: \[ \frac{b}{a} = \frac{c}{b} \quad \Rightarrow \quad b^2 = ac. \quad \cdots (1) \]

Step 2: Taking Logarithms
Now, let's take the logarithm of both sides of equation (1): \[ \log b^2 = \log (ac). \] Using the logarithmic property \( \log(xy) = \log x + \log y \), we get: \[ 2 \log b = \log a + \log c. \quad \cdots (2) \] This equation shows that the logarithms of the terms, \( \log a, \log b, \log c \), satisfy the condition for being in an arithmetic progression, because in an arithmetic progression, the middle term is the average of the first and third terms.

Step 3: Verifying Arithmetic Progression
For terms to be in arithmetic progression, the middle term should satisfy: \[ 2 \log b = \log a + \log c. \] This is exactly what we obtained in step 2, confirming that \( \log a, \log b, \log c \) are in arithmetic progression.