# Let a, b, c be three numbers in G. P . And we take theirs log (eg. Log a, Log b,Log c) is it always true that their log will be in A. P?

Aug 4, 2018

Please refer to a Proof in the Explanation.

#### Explanation:

$a , b , c$ are in GP.

$\therefore \text{The common ratio} = \frac{b}{a} = \frac{c}{b}$.

Clearly, ${b}^{2} = a c$.

$L o g {b}^{2} = L o g \left(a c\right)$.

$\therefore 2 L o g b = L o g a + L o g c$.

Consequently, $\log a , \log b \mathmr{and} \log c$ are in AP.