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Question Number 25315 by ibraheem160 last updated on 08/Dec/17

$$if{log}_a^b={log}_b^c={log}_c^a.showthata=b=c$$

Answered by prakash jain last updated on 08/Dec/17

$${\log }_{a}b{\log }_{b}c={\log }_{a}c$$$$\Rightarrow {\left({\log }_{c}a\right)}^2={\log }_{a}c$$$$\Rightarrow 1={\left({\log }_{a}c\right)}^3\Rightarrow {\log }_{a}c=1$$$$\Rightarrow a=c$$$${\log }_{c}a={\log }_{b}c={\log }_{a}b=1$$$$\Rightarrow a=b=c$$

Answered by Rasheed.Sindhi last updated on 08/Dec/17

$$\mathrm{Let}{\log }_{\mathrm{a}}\mathrm{b}={\log }_{\mathrm{b}}\mathrm{c}={\log }_{\mathrm{c}}\mathrm{a}=\mathrm{y}$$$${\mathrm{a}}^{\mathrm{y}}=\mathrm{b},{\mathrm{b}}^{\mathrm{y}}=\mathrm{c},{\mathrm{c}}^{\mathrm{y}}=\mathrm{a}$$$${\mathrm{a}}^{\mathrm{y}}=\mathrm{b},{\mathrm{b}}^{\mathrm{y}}=\mathrm{c}\Rightarrow {\left({\mathrm{a}}^{\mathrm{y}}\right)}^{\mathrm{y}}=\mathrm{c}$$$${\mathrm{c}}^{\mathrm{y}}=\mathrm{a},{\left({\mathrm{a}}^{\mathrm{y}}\right)}^{\mathrm{y}}=\mathrm{c}\Rightarrow {\left\{{\left({\mathrm{c}}^{\mathrm{y}}\right)}^{\mathrm{y}}\right\}}^{\mathrm{y}}=\mathrm{c}$$$${\mathrm{c}}^{{\mathrm{y}}^3}=\mathrm{c}\Rightarrow {\mathrm{y}}^3=1\Rightarrow \mathrm{y}=1$$$${\mathrm{a}}^{\mathrm{y}}=\mathrm{b}\Rightarrow {\mathrm{a}}^1=\mathrm{b}\Rightarrow \mathrm{a}=\mathrm{b}$$$${\mathrm{b}}^{\mathrm{y}}=\mathrm{c}\Rightarrow {\mathrm{b}}^1=\mathrm{c}\Rightarrow \mathrm{b}=\mathrm{c}$$$$\mathrm{a}=\mathrm{b},\mathrm{b}=\mathrm{c}\Rightarrow \mathrm{a}=\mathrm{b}=\mathrm{c}$$

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