If a, b, and c are in G.P., prove that
log a, log b, and log c are in A.P.

Given: If a, b, and c are in G.P.

To do: To prove that log a, log b and log c are in A.P.

Answer:

a, b, and c are in G.P ⟹ $\frac{c}{b} = \frac{b}{a}$ or

$b^2 = ac$

Now taking logs on both sides

$log b^2 = log ac$

Using properties of logs like

$log x^m = m \times logx \ and \ logxy = log x + log y$

$log \ b^2 = log \ ac$

⟹$2 log \ b = log \ a + log \ c$

or $log \ c - log \ b = log \ b - log \ a$  which

⟹$log \ a, log \ b, and log \ c$ are in AP by definition.

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Updated on: 10-Oct-2022

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