If log a, log b, log c are in arithmetic progression (A.P.), what can be said about a, b, c?
Aptitude
Logarithm
Difficulty: Easy
Choose an option
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Aa, b, c are in G.P.
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Ba^2, b^2, c^2 are in G.P.
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Ca, b, c are in A.P.
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DNone of these
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E—
Answer
Correct Answer: a, b, c are in G.P.
Explanation
Introduction / Context:Relating progressions of logarithms to progressions of the original numbers is a standard property: an A.P. in logs corresponds to a G.P. in the numbers because the log function converts products into sums.
Given Data / Assumptions:
- log a, log b, log c are in A.P. ⇒ 2 log b = log a + log c.
- a, b, c > 0.
Concept / Approach:
- Use log rules: log a + log c = log(ac).
- Equate and exponentiate if needed.
Step-by-Step Reasoning:
2 log b = log a + log c ⇒ log b^2 = log(ac)Therefore, b^2 = ac ⇒ b is the geometric mean of a and cHence a, b, c are in geometric progression (G.P.).Verification / Alternative check:Example: a = 2, c = 8 ⇒ b = √(16) = 4; then log 2, log 4, log 8 differ by equal amounts; numbers 2, 4, 8 are in G.P.
Why Other Options Are Wrong:
- Squares in G.P. is true if and only if a, b, c are in G.P., but the direct, standard statement is that a, b, c themselves are in G.P.
- A.P. for a, b, c contradicts multiplicative spacing indicated by logs in A.P.
Final Answer:a, b, c are in G.P.