If log a, log b, log c are in arithmetic progression (A.P.), what can be said about a, b, c?

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:

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.