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Question Number 177360 by jlewis last updated on 04/Oct/22

1/2 log_4 36 ×log_6 64

$$\mathrm{1}/\mathrm{2}\:{log}_{\mathrm{4}} \mathrm{36}\:×{log}_{\mathrm{6}} \mathrm{64} \\ $$

Answered by TheHoneyCat last updated on 08/Oct/22

=(1/2)log_4 (4×9)×log_6 (6×9) =(1/2)(log_4 4+log_4 9)(log_6 6+log_6 9) =(1/2)(1+((ln9)/(ln4)))(1+((ln9)/(ln6))) =(1/2)(1+((2 ln3)/(2 ln2)))(1+((2 ln3)/(ln2+ln3))) =(1/2)(1+((ln3)/(ln2)))(1+((2 ln3)/(ln2+ln3))) =(1/2) ×((ln2 + ln3)/(ln2))×((ln2 + 3 ln3)/(ln2+ln3)) =((ln2 + 3 ln3)/(2 ln2)) _□

$$=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{log}_{\mathrm{4}} \left(\mathrm{4}×\mathrm{9}\right)×\mathrm{log}_{\mathrm{6}} \left(\mathrm{6}×\mathrm{9}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{log}_{\mathrm{4}} \mathrm{4}+\mathrm{log}_{\mathrm{4}} \mathrm{9}\right)\left(\mathrm{log}_{\mathrm{6}} \mathrm{6}+\mathrm{log}_{\mathrm{6}} \mathrm{9}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+\frac{\mathrm{ln9}}{\mathrm{ln4}}\right)\left(\mathrm{1}+\frac{\mathrm{ln9}}{\mathrm{ln6}}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+\frac{\mathrm{2}\:\mathrm{ln3}}{\mathrm{2}\:\mathrm{ln2}}\right)\left(\mathrm{1}+\frac{\mathrm{2}\:\mathrm{ln3}}{\mathrm{ln2}+\mathrm{ln3}}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+\frac{\mathrm{ln3}}{\mathrm{ln2}}\right)\left(\mathrm{1}+\frac{\mathrm{2}\:\mathrm{ln3}}{\mathrm{ln2}+\mathrm{ln3}}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\:×\frac{\mathrm{ln2}\:+\:\mathrm{ln3}}{\mathrm{ln2}}×\frac{\mathrm{ln2}\:+\:\mathrm{3}\:\mathrm{ln3}}{\mathrm{ln2}+\mathrm{ln3}} \\ $$$$=\frac{\mathrm{ln2}\:+\:\mathrm{3}\:\mathrm{ln3}}{\mathrm{2}\:\mathrm{ln2}}\:_{\Box} \\ $$

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