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Question Number 17779 by Mr easymsn last updated on 10/Jul/17

$$showthat\left\{lo\underset{a}{gab}\right\}\left\{lo\underset{b}{gab}\right\}=loga\underset{a}{b}+loga\underset{b}{b}$$

Answered by Tinkutara last updated on 11/Jul/17

$$\left({\log }_{a}ab\right)\left({\log }_{b}ab\right)$$$$=(1+{\log }_{a}b)(1+{\log }_{b}a)$$$$=1+{\log }_{b}a+{\log }_{a}b+1$$$$={\log }_{b}b+{\log }_{b}a+{\log }_{a}b+{\log }_{a}a$$$$={\mathbf{log}}_{\mathit{a}}\mathit{a}\mathit{b}+{\mathbf{log}}_{\mathit{b}}\mathit{a}\mathit{b}$$

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