Question

If three numbers be in G.P, then their logarithms will be in

• A. A.P
• B. G.P
• C. H.P
• D. None of these

Answer 1

The correct option is A

A.P

Explanation for the correct option:

Finding the relation:

Let's say $\mathrm{a},\mathrm{b},\mathrm{c}$ are in G.P

According to the properties of Geometric progression, three non-zero terms $\mathrm{a},\mathrm{b},\mathrm{c}$ are in G.P if and only if ,

$\Rightarrow {\mathrm{b}}^2=\mathrm{ac}$

Taking $\log$ on both sides we get,

$$\begin{gathered} \log \left({\mathrm{b}}^2\right)=\log \left(\mathrm{ac}\right) \\ \Rightarrow 2\mathrm{logb}=\mathrm{loga}+\mathrm{logc}\left[\because {\mathrm{loga}}^b=\mathrm{bloga},\log \left(\mathrm{ab}\right)=\mathrm{loga}+\mathrm{logb}\right] \\ \Rightarrow \mathrm{logb}=\frac{\mathrm{loga}+\mathrm{logc}}{2} \end{gathered}$$

$\therefore \mathrm{loga},\mathrm{logb},\mathrm{logc}$ are in A.P.

Hence, the correct option is A.