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Question Number 139456 by SOMEDAVONG last updated on 27/Apr/21

$$\mathrm{E}={\log }_{\mathrm{a}}\sqrt{\frac{\mathrm{c}}{{\mathrm{b}}^{{\log }_{\mathrm{ab}}\mathrm{c}}}}+{\log }_{\mathrm{b}}\sqrt{\frac{\mathrm{c}}{{\mathrm{a}}^{{\log }_{\mathrm{ab}}\mathrm{c}}}}$$

Answered by bemath last updated on 27/Apr/21

$$\mathrm{E}=\log {}_{\mathrm{a}}\sqrt{\frac{\mathrm{c}}{{\mathrm{c}}^{\log {}_{\mathrm{ab}}\mathrm{b}}}}+\log {}_{\mathrm{b}}\sqrt{\frac{\mathrm{c}}{{\mathrm{c}}^{\log {}_{\mathrm{ab}}\mathrm{a}}}}$$$$\mathrm{E}=\frac{1}{2}[\log {}_{\mathrm{a}}\left({\mathrm{c}}^{1-\log {}_{\mathrm{ab}}\mathrm{b}}\right)+\log {}_{\mathrm{b}}\left({\mathrm{c}}^{1-\log {}_{\mathrm{ab}}\mathrm{a}}\right)]$$$$\mathrm{E}=\frac{1}{2}[\log {}_{\mathrm{a}}\left({\mathrm{c}}^{\log {}_{\mathrm{ab}}\left(\frac{1}{\mathrm{a}}\right)}\right)+\log {}_{\mathrm{b}}\left({\mathrm{c}}^{\log {}_{\mathrm{ab}}\left(\frac{1}{\mathrm{b}}\right)}\right)]$$$$$$

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