Question Number 141469 by otchereabdullai@gmail.com last updated on 19/May/21
$$\mathrm{simplify} \\ $$$$\left(\mathrm{log}_{\mathrm{3}} \mathrm{11}\right)\left(\mathrm{log}_{\mathrm{11}} \mathrm{13}\right)\left(\mathrm{log}_{\mathrm{13}} \mathrm{15}\right)\left(\mathrm{log}_{\mathrm{15}} \mathrm{27}\right)\left(\mathrm{log}_{\mathrm{27}} \mathrm{81}\right) \\ $$$$\mathrm{please}\:\mathrm{i}\:\mathrm{need}\:\mathrm{help} \\ $$
Answered by bramlexs22 last updated on 19/May/21
$$\:\mathrm{log}\:_{\mathrm{3}} \mathrm{11}.\:\mathrm{log}\:_{\mathrm{11}} \:\mathrm{13}\:=\:\mathrm{log}\:_{\mathrm{3}} \mathrm{13} \\ $$$$\mathrm{log}\:_{\mathrm{13}} \mathrm{15}.\mathrm{log}\:_{\mathrm{15}} \mathrm{27}\:=\:\mathrm{log}\:_{\mathrm{13}} \mathrm{27} \\ $$$$\Rightarrow\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{13}\right).\mathrm{log}\:_{\mathrm{13}} \left(\mathrm{27}\right)=\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{27}\right) \\ $$$$\Rightarrow\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{27}\right).\mathrm{log}\:_{\mathrm{27}} \left(\mathrm{81}\right)=\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{81}\right)=\mathrm{4} \\ $$
Commented by otchereabdullai@gmail.com last updated on 19/May/21
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}! \\ $$
Answered by Rasheed.Sindhi last updated on 19/May/21
$$\frac{\cancel{\mathrm{log11}}}{\mathrm{log3}}.\frac{\cancel{\mathrm{log13}}}{\cancel{\mathrm{log11}}}.\frac{\cancel{\mathrm{log15}}}{\cancel{\mathrm{log13}}}.\frac{\cancel{\mathrm{log27}}}{\cancel{\mathrm{log15}}}.\frac{\mathrm{log81}}{\cancel{\mathrm{log27}}} \\ $$$$=\frac{\mathrm{log81}}{\mathrm{log3}}=\mathrm{log}_{\mathrm{3}} \mathrm{81}=\mathrm{4} \\ $$
Commented by otchereabdullai@gmail.com last updated on 19/May/21
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}! \\ $$
Answered by mr W last updated on 19/May/21
$$\mathrm{log}_{{a}} \:{b}=\frac{\mathrm{ln}\:{b}}{\mathrm{ln}\:{a}} \\ $$$$\left(\mathrm{log}_{\mathrm{3}} \mathrm{11}\right)\left(\mathrm{log}_{\mathrm{11}} \mathrm{13}\right)\left(\mathrm{log}_{\mathrm{13}} \mathrm{15}\right)\left(\mathrm{log}_{\mathrm{15}} \mathrm{27}\right)\left(\mathrm{log}_{\mathrm{27}} \mathrm{81}\right) \\ $$$$=\frac{\mathrm{ln}\:\mathrm{11}}{\mathrm{ln}\:\mathrm{3}}×\frac{\mathrm{ln}\:\mathrm{13}}{\mathrm{ln}\:\mathrm{11}}×\frac{\mathrm{ln}\:\mathrm{15}}{\mathrm{ln}\:\mathrm{13}}×\frac{\mathrm{ln}\:\mathrm{27}}{\mathrm{ln}\:\mathrm{15}}×\frac{\mathrm{ln}\:\mathrm{81}}{\mathrm{ln}\:\mathrm{27}} \\ $$$$=\frac{\mathrm{ln}\:\mathrm{81}}{\mathrm{ln}\:\mathrm{3}} \\ $$$$=\frac{\mathrm{ln}\:\mathrm{3}^{\mathrm{4}} }{\mathrm{ln}\:\mathrm{3}} \\ $$$$=\frac{\mathrm{4}×\mathrm{ln}\:\mathrm{3}}{\mathrm{ln}\:\mathrm{3}} \\ $$$$=\mathrm{4} \\ $$
Commented by otchereabdullai@gmail.com last updated on 19/May/21
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{prof}\: \\ $$