Question Number 68721 by ahmadshah last updated on 15/Sep/19
Commented by Prithwish sen last updated on 15/Sep/19
$$\mathrm{3} \\ $$
Answered by $@ty@m123 last updated on 15/Sep/19
$$\mathrm{log}\:_{\mathrm{2}} \mathrm{3}×\mathrm{log}\:_{\mathrm{3}} \mathrm{4}×\mathrm{log}\:_{\mathrm{4}} \mathrm{5}×\mathrm{log}\:_{\mathrm{5}} \mathrm{6}×\mathrm{log}\:_{\mathrm{6}} \mathrm{7}×\mathrm{log}\:_{\mathrm{7}} \mathrm{8} \\ $$$$=\frac{\mathrm{log}_{\mathrm{3}} \:\mathrm{4}}{\mathrm{log}_{\mathrm{3}} \:\mathrm{2}}×\frac{\mathrm{log}_{\mathrm{5}} \:\mathrm{6}}{\mathrm{log}_{\mathrm{5}} \mathrm{4}\:}×\frac{\mathrm{log}_{\mathrm{7}} \:\mathrm{8}}{\mathrm{log}_{\mathrm{7}} \:\mathrm{6}} \\ $$$$=\mathrm{log}_{\mathrm{2}} \:\mathrm{4}×\mathrm{log}_{\mathrm{4}} \:\mathrm{6}×\mathrm{log}_{\mathrm{6}} \:\mathrm{8} \\ $$$$=\mathrm{log}_{\mathrm{2}} \:\mathrm{4}×\frac{\mathrm{log}_{\mathrm{6}} \:\mathrm{8}}{\mathrm{log}_{\mathrm{6}} \:\mathrm{4}} \\ $$$$=\mathrm{log}_{\mathrm{2}} \:\mathrm{4}×\mathrm{log}_{\mathrm{4}} \:\mathrm{8} \\ $$$$=\mathrm{log}_{\mathrm{2}} \:\mathrm{4}×\mathrm{log}_{\mathrm{4}} \:\mathrm{2}^{\mathrm{3}} \\ $$$$=\mathrm{log}_{\mathrm{2}} \:\mathrm{4}×\mathrm{3log}_{\mathrm{4}} \:\mathrm{2} \\ $$$$=\mathrm{log}_{\mathrm{2}} \:\mathrm{4}×\frac{\mathrm{3}}{\mathrm{log}_{\mathrm{2}} \:\mathrm{4}} \\ $$$$=\mathrm{3} \\ $$