Question

If log_xa, a^x/2 and log_b x are in G.P., then write the value of x.

Answer 1

$$\begin{gathered} {\log }_{x}a\mathit{,}\mathit{}{a}^{\frac{x}{2}}\mathit{}\mathrm{and}\mathit{}{\mathit{log}}_{b}x\mathit{}\mathrm{are}\mathrm{in}\mathrm{G}.\mathrm{P}. \\ \therefore {\left({\mathrm{a}}^{\frac{x}{2}}\right)}^2={\log }_x\mathrm{a}\times {\mathit{log}}_{\mathrm{b}}x\mathit{} \\ \mathit{\Rightarrow }{a}^x\mathit{}\mathit{=}\frac{\mathit{}{\mathit{log}}_{b}a\mathit{}}{{\mathit{log}}_{b}x}\times \mathit{}\mathit{}{\mathit{log}}_{b}x\mathit{} \\ \mathit{\Rightarrow }{a}^x\mathit{}\mathit{=}\mathit{}\mathit{}{\mathit{log}}_{b}a\mathit{} \\ \mathrm{Now},\mathrm{by}\mathrm{taking}{\log }_{\mathrm{a}}\mathrm{on}\mathrm{both}\mathrm{the}\mathrm{sides}: \\ \mathit{\Rightarrow }\mathrm{x}={\log }_{\mathrm{a}}\left({\mathit{log}}_{\mathrm{b}}\mathrm{a}\right) \end{gathered}$$