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Question Number 68721 by ahmadshah last updated on 15/Sep/19

Commented by Prithwish sen last updated on 15/Sep/19

3

$$\mathrm{3} \\ $$

Answered by $@ty@m123 last updated on 15/Sep/19

log _2 3×log _3 4×log _4 5×log _5 6×log _6 7×log _7 8  =((log_3  4)/(log_3  2))×((log_5  6)/(log_5 4 ))×((log_7  8)/(log_7  6))  =log_2  4×log_4  6×log_6  8  =log_2  4×((log_6  8)/(log_6  4))  =log_2  4×log_4  8  =log_2  4×log_4  2^3   =log_2  4×3log_4  2  =log_2  4×(3/(log_2  4))  =3

$$\mathrm{log}\:_{\mathrm{2}} \mathrm{3}×\mathrm{log}\:_{\mathrm{3}} \mathrm{4}×\mathrm{log}\:_{\mathrm{4}} \mathrm{5}×\mathrm{log}\:_{\mathrm{5}} \mathrm{6}×\mathrm{log}\:_{\mathrm{6}} \mathrm{7}×\mathrm{log}\:_{\mathrm{7}} \mathrm{8} \\ $$$$=\frac{\mathrm{log}_{\mathrm{3}} \:\mathrm{4}}{\mathrm{log}_{\mathrm{3}} \:\mathrm{2}}×\frac{\mathrm{log}_{\mathrm{5}} \:\mathrm{6}}{\mathrm{log}_{\mathrm{5}} \mathrm{4}\:}×\frac{\mathrm{log}_{\mathrm{7}} \:\mathrm{8}}{\mathrm{log}_{\mathrm{7}} \:\mathrm{6}} \\ $$$$=\mathrm{log}_{\mathrm{2}} \:\mathrm{4}×\mathrm{log}_{\mathrm{4}} \:\mathrm{6}×\mathrm{log}_{\mathrm{6}} \:\mathrm{8} \\ $$$$=\mathrm{log}_{\mathrm{2}} \:\mathrm{4}×\frac{\mathrm{log}_{\mathrm{6}} \:\mathrm{8}}{\mathrm{log}_{\mathrm{6}} \:\mathrm{4}} \\ $$$$=\mathrm{log}_{\mathrm{2}} \:\mathrm{4}×\mathrm{log}_{\mathrm{4}} \:\mathrm{8} \\ $$$$=\mathrm{log}_{\mathrm{2}} \:\mathrm{4}×\mathrm{log}_{\mathrm{4}} \:\mathrm{2}^{\mathrm{3}} \\ $$$$=\mathrm{log}_{\mathrm{2}} \:\mathrm{4}×\mathrm{3log}_{\mathrm{4}} \:\mathrm{2} \\ $$$$=\mathrm{log}_{\mathrm{2}} \:\mathrm{4}×\frac{\mathrm{3}}{\mathrm{log}_{\mathrm{2}} \:\mathrm{4}} \\ $$$$=\mathrm{3} \\ $$

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