Question

If a, b, c are in G.P., prove that
$\frac{1}{{\log }_{a}m},\frac{1}{{\log }_{b}m},\frac{1}{{\log }_{c}m}$ are in A.P.

Answer 1

a, b, c are in G.P.
$$\begin{gathered} \therefore b^2=ac \\ \mathrm{Now}\mathrm{taking}\mathit{}lo{g}_m\mathrm{on}\mathrm{both}\mathrm{the}\mathrm{sides}: \\ \Rightarrow lo{g}_m{\left(b\right)}^{\mathit{2}}\mathit{}\mathit{=}\mathit{}lo{g}_m\left(ac\right) \\ \mathit{\Rightarrow }\mathit{2}lo{g}_m\left(b\right)\mathit{}\mathit{=}\mathit{}lo{g}_{m}a\mathit{}\mathit{}\mathit{+}\mathit{}lo{g}_m\left(c\right) \\ \Rightarrow \frac{2}{{\log }_b\left(m\right)}=\frac{1}{{\log }_a\left(m\right)}+\frac{1}{{\log }_c\left(m\right)} \\ \mathrm{Thus},\frac{1}{{\log }_{\mathrm{a}}\left(\mathrm{m}\right)},\frac{1}{{\log }_{\mathrm{b}}\left(\mathrm{m}\right)}\mathrm{and}\frac{1}{{\log }_{\mathrm{c}}\left(\mathrm{m}\right)}\mathrm{are}\mathrm{in}\mathrm{A}.\mathrm{P}. \end{gathered}$$