If a, b, c be in GP, then log aⁿ, log b ⁿ, log c ⁿ will be

If a,b,c, are in GP, then b² = ac

Taking  log on  both sides,  we  get

$\begin{array}{l}2{\log }_{\mathrm{e}}\mathrm{b}={\log }_{\mathrm{e}}\mathrm{a}+{\log }_{\mathrm{e}}\mathrm{c}\\ \Rightarrow 2{\mathrm{nlog}}_{\mathrm{e}}\mathrm{b}={\mathrm{nlog}}_{\mathrm{e}}\mathrm{a}+{\mathrm{nlog}}_{\mathrm{e}}\mathrm{c}\\ \Rightarrow 2{\log }_{\mathrm{e}}{\mathrm{b}}^{\mathrm{n}}={\log }_{\mathrm{e}}{\mathrm{a}}^{\mathrm{n}}+{\log }_{\mathrm{e}}{\mathrm{c}}^{\mathrm{n}}\\ \Rightarrow {\log }_{\mathrm{c}}{\mathrm{a}}^{\mathrm{n}},{\log }_{\mathrm{e}}{\mathrm{b}}^{\mathrm{n}},{\log }_{\mathrm{e}}{\mathrm{c}}^{\mathrm{n}}\text{ be in AP. }\end{array}$