Question

If $a,b,c$ are in $G.P$ then ${\log }_a\left(x\right),{\log }_b\left(x\right),{\log }_c\left(x\right)$ are in

Answer 1

Find the value of $\mathit{l}\mathit{o}{\mathit{g}}_{\mathbf{a}}\left(x\right)\mathbf{,}\mathbf{}\mathit{l}\mathit{o}{\mathit{g}}_{\mathbf{b}}\left(x\right)\mathbf{,}\mathbf{}\mathit{l}\mathit{o}{\mathit{g}}_{\mathbf{c}}\left(x\right)$:

Given $a,b,c$ are in $G.P$

$$\begin{gathered} \Rightarrow b^2=ac \\ \Rightarrow 2\log b=\log a+\log c \\ \Rightarrow 2\frac{\log b}{\log x}=\frac{\log a}{\log x}+\frac{\log c}{\log x} \\ \Rightarrow 2{\log }_{x}b={\log }_{x}a+{\log }_{x}c \\ \Rightarrow \frac{2}{{\log }_{b}x}=\frac{1}{{\log }_{a}x}+\frac{1}{{\log }_{c}x} \end{gathered}$$

So, the given terms are in $H.P$

Hence, the terms $\mathit{l}\mathit{o}{\mathit{g}}_{\mathbf{a}}\mathit{x}\mathbf{,}\mathbf{}\mathit{l}\mathit{o}{\mathit{g}}_{\mathbf{b}}\mathit{x}\mathbf{,}\mathbf{}\mathit{l}\mathit{o}{\mathit{g}}_{\mathbf{c}}\mathit{x}\mathbf{}$ are in $\mathit{H}\mathbf{.}\mathit{P}\mathbf{.}$