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

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

Answered by bramlexs22 last updated on 19/May/21

$$\log {}_{3}11.\log {}_{11}13=\log {}_{3}13$$$$\log {}_{13}15.\log {}_{15}27=\log {}_{13}27$$$$\Rightarrow \log {}_3\left(13\right).\log {}_{13}\left(27\right)=\log {}_3\left(27\right)$$$$\Rightarrow \log {}_3\left(27\right).\log {}_{27}\left(81\right)=\log {}_3\left(81\right)=4$$

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

$$\frac{\cancel{\log 11}}{\log 3}.\frac{\cancel{\log 13}}{\cancel{\log 11}}.\frac{\cancel{\log 15}}{\cancel{\log 13}}.\frac{\cancel{\log 27}}{\cancel{\log 15}}.\frac{\log 81}{\cancel{\log 27}}$$$$=\frac{\log 81}{\log 3}={\log }_{3}81=4$$

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

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

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

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

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