If log_ab=2 ; log_bc=2 and log₃c=3+log₃a then (a+b+c) equals .... ?

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$$\begin{gathered} {\log }_{\mathrm{a}}\left(\mathrm{b}\right)=2 \\ \Rightarrow \mathrm{b}={\mathrm{a}}^2 ;\mathrm{equation}\left(\mathrm{i}\right) \\ {\log }_{\mathrm{b}}\left(\mathrm{c}\right)=2 \\ \Rightarrow \mathrm{c}={\mathrm{b}}^2 ;\mathrm{equation}\left(\mathrm{ii}\right) \\ {\log }_3\left(\mathrm{c}\right)=3+{\log }_3\left(\mathrm{a}\right) \\ \Rightarrow {\log }_3\left(\mathrm{c}\right)-{\log }_3\left(\mathrm{a}\right)=3 \\ \mathrm{Using} \mathrm{identity} {\log }_{\mathrm{p}}\left(\mathrm{x}\right)-{\log }_{\mathrm{p}}\left(\mathrm{y}\right)={\log }_{\mathrm{p}}\left(\frac{\mathrm{x}}{\mathrm{y}}\right) , \mathrm{we} \mathrm{get}, \\ {\log }_3\left(\frac{\mathrm{c}}{\mathrm{a}}\right)=3 \\ \Rightarrow \frac{\mathrm{c}}{\mathrm{a}}=3^3 \\ \Rightarrow \frac{\mathrm{c}}{\mathrm{a}}=27 \\ \Rightarrow \mathrm{c}=27\mathrm{a} \\ \Rightarrow {\mathrm{b}}^2=27\mathrm{a} \left\{\mathrm{By} \mathrm{equation} \left(\mathrm{ii}\right)\right\} \\ \Rightarrow {\left({\mathrm{a}}^2\right)}^2=27\mathrm{a} \left\{\mathrm{By} \mathrm{equation} \left(\mathrm{ii}\right)\right\} \\ \Rightarrow {\mathrm{a}}^4-27\mathrm{a} \\ \Rightarrow \mathrm{a}\left({\mathrm{a}}^3-27\right)=0 \\ \mathrm{But} \mathrm{a}\ne 0 \mathrm{as} \log \mathrm{will} \mathrm{be} \mathrm{undefined}. \\ \Rightarrow {\mathrm{a}}^3-27=0 \\ \Rightarrow {\mathrm{a}}^3=27 \\ \Rightarrow \mathrm{a}=3 \\ \mathrm{Putting} \mathrm{this} \mathrm{value} \mathrm{in} \mathrm{equation}\left(\mathrm{i}\right), \mathrm{we} \mathrm{get} \\ \mathrm{b}=3^2 \\ \Rightarrow \mathrm{b}=9 \\ \mathrm{Putting} \mathrm{this} \mathrm{value} \mathrm{in} \mathrm{equation}\left(\mathrm{ii}\right), \mathrm{we} \mathrm{get} \\ \mathrm{c}=9^2 \\ \Rightarrow \mathrm{c}=81 \\ \Rightarrow \mathrm{a}+\mathrm{b}+\mathrm{c}=3+9+81 \\ \mathbf{\Rightarrow }\mathbf{a}\mathbf{+}\mathbf{b}\mathbf{+}\mathbf{c}\mathbf{=}\mathbf{93} \end{gathered}$$

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