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Question Number 141469 by otchereabdullai@gmail.com last updated on 19/May/21

simplify (log_3 11)(log_(11) 13)(log_(13) 15)(log_(15) 27)(log_(27) 81) please i need help

$$\mathrm{simplify} \\ $$$$\left(\mathrm{log}_{\mathrm{3}} \mathrm{11}\right)\left(\mathrm{log}_{\mathrm{11}} \mathrm{13}\right)\left(\mathrm{log}_{\mathrm{13}} \mathrm{15}\right)\left(\mathrm{log}_{\mathrm{15}} \mathrm{27}\right)\left(\mathrm{log}_{\mathrm{27}} \mathrm{81}\right) \\ $$$$\mathrm{please}\:\mathrm{i}\:\mathrm{need}\:\mathrm{help} \\ $$

Answered by bramlexs22 last updated on 19/May/21

log _3 11. log _(11) 13 = log _3 13 log _(13) 15.log _(15) 27 = log _(13) 27 ⇒log _3 (13).log _(13) (27)=log _3 (27) ⇒log _3 (27).log _(27) (81)=log _3 (81)=4

$$\:\mathrm{log}\:_{\mathrm{3}} \mathrm{11}.\:\mathrm{log}\:_{\mathrm{11}} \:\mathrm{13}\:=\:\mathrm{log}\:_{\mathrm{3}} \mathrm{13} \\ $$$$\mathrm{log}\:_{\mathrm{13}} \mathrm{15}.\mathrm{log}\:_{\mathrm{15}} \mathrm{27}\:=\:\mathrm{log}\:_{\mathrm{13}} \mathrm{27} \\ $$$$\Rightarrow\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{13}\right).\mathrm{log}\:_{\mathrm{13}} \left(\mathrm{27}\right)=\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{27}\right) \\ $$$$\Rightarrow\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{27}\right).\mathrm{log}\:_{\mathrm{27}} \left(\mathrm{81}\right)=\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{81}\right)=\mathrm{4} \\ $$

Commented by otchereabdullai@gmail.com last updated on 19/May/21

God bless you sir!

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}! \\ $$

Answered by Rasheed.Sindhi last updated on 19/May/21

((log11)/(log3)).((log13)/(log11)).((log15)/(log13)).((log27)/(log15)).((log81)/(log27)) =((log81)/(log3))=log_3 81=4

$$\frac{\cancel{\mathrm{log11}}}{\mathrm{log3}}.\frac{\cancel{\mathrm{log13}}}{\cancel{\mathrm{log11}}}.\frac{\cancel{\mathrm{log15}}}{\cancel{\mathrm{log13}}}.\frac{\cancel{\mathrm{log27}}}{\cancel{\mathrm{log15}}}.\frac{\mathrm{log81}}{\cancel{\mathrm{log27}}} \\ $$$$=\frac{\mathrm{log81}}{\mathrm{log3}}=\mathrm{log}_{\mathrm{3}} \mathrm{81}=\mathrm{4} \\ $$

Commented by otchereabdullai@gmail.com last updated on 19/May/21

God bless you sir!

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}! \\ $$

Answered by mr W last updated on 19/May/21

log_a b=((ln b)/(ln a)) (log_3 11)(log_(11) 13)(log_(13) 15)(log_(15) 27)(log_(27) 81) =((ln 11)/(ln 3))×((ln 13)/(ln 11))×((ln 15)/(ln 13))×((ln 27)/(ln 15))×((ln 81)/(ln 27)) =((ln 81)/(ln 3)) =((ln 3^4 )/(ln 3)) =((4×ln 3)/(ln 3)) =4

$$\mathrm{log}_{{a}} \:{b}=\frac{\mathrm{ln}\:{b}}{\mathrm{ln}\:{a}} \\ $$$$\left(\mathrm{log}_{\mathrm{3}} \mathrm{11}\right)\left(\mathrm{log}_{\mathrm{11}} \mathrm{13}\right)\left(\mathrm{log}_{\mathrm{13}} \mathrm{15}\right)\left(\mathrm{log}_{\mathrm{15}} \mathrm{27}\right)\left(\mathrm{log}_{\mathrm{27}} \mathrm{81}\right) \\ $$$$=\frac{\mathrm{ln}\:\mathrm{11}}{\mathrm{ln}\:\mathrm{3}}×\frac{\mathrm{ln}\:\mathrm{13}}{\mathrm{ln}\:\mathrm{11}}×\frac{\mathrm{ln}\:\mathrm{15}}{\mathrm{ln}\:\mathrm{13}}×\frac{\mathrm{ln}\:\mathrm{27}}{\mathrm{ln}\:\mathrm{15}}×\frac{\mathrm{ln}\:\mathrm{81}}{\mathrm{ln}\:\mathrm{27}} \\ $$$$=\frac{\mathrm{ln}\:\mathrm{81}}{\mathrm{ln}\:\mathrm{3}} \\ $$$$=\frac{\mathrm{ln}\:\mathrm{3}^{\mathrm{4}} }{\mathrm{ln}\:\mathrm{3}} \\ $$$$=\frac{\mathrm{4}×\mathrm{ln}\:\mathrm{3}}{\mathrm{ln}\:\mathrm{3}} \\ $$$$=\mathrm{4} \\ $$

Commented by otchereabdullai@gmail.com last updated on 19/May/21

God bless you prof

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{prof}\: \\ $$

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