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Question Number 65972 by gunawan last updated on 07/Aug/19

If ((log_2  a)/(log_3  b))=m and ((log_3  a)/(log_2  b))=n  a>1 and b>1  then (m/n)=...  a.log_2  3  b. log_3  2  c. log_4  9  d. (log_2  3)^2   e. (log_3  2)^2

$$\mathrm{If}\:\frac{\mathrm{log}_{\mathrm{2}} \:{a}}{\mathrm{log}_{\mathrm{3}} \:{b}}={m}\:\mathrm{and}\:\frac{\mathrm{log}_{\mathrm{3}} \:{a}}{\mathrm{log}_{\mathrm{2}} \:{b}}={n} \\ $$ $${a}>\mathrm{1}\:\mathrm{and}\:{b}>\mathrm{1} \\ $$ $$\mathrm{then}\:\frac{{m}}{{n}}=... \\ $$ $${a}.\mathrm{log}_{\mathrm{2}} \:\mathrm{3} \\ $$ $${b}.\:\mathrm{log}_{\mathrm{3}} \:\mathrm{2} \\ $$ $${c}.\:\mathrm{log}_{\mathrm{4}} \:\mathrm{9} \\ $$ $${d}.\:\left(\mathrm{log}_{\mathrm{2}} \:\mathrm{3}\right)^{\mathrm{2}} \\ $$ $${e}.\:\left(\mathrm{log}_{\mathrm{3}} \:\mathrm{2}\right)^{\mathrm{2}} \\ $$

Answered by MJS last updated on 07/Aug/19

m=(((ln a)/(ln 2))/((ln b)/(ln 3)))=((ln 3 ln a)/(ln 2 ln b)); n=(((ln a)/(ln 3))/((ln b)/(ln 2)))=((ln 2 ln a)/(ln 3 ln b))  (m/n)=(((ln 3 ln a)/(ln 2 ln b))/((ln 2 ln a)/(ln 3 ln b)))=(((ln 3)^2 )/((ln 2)^2 ))=(((ln 3)/(ln 2)))^2 =(log_2  3)^2

$${m}=\frac{\frac{\mathrm{ln}\:{a}}{\mathrm{ln}\:\mathrm{2}}}{\frac{\mathrm{ln}\:{b}}{\mathrm{ln}\:\mathrm{3}}}=\frac{\mathrm{ln}\:\mathrm{3}\:\mathrm{ln}\:{a}}{\mathrm{ln}\:\mathrm{2}\:\mathrm{ln}\:{b}};\:{n}=\frac{\frac{\mathrm{ln}\:{a}}{\mathrm{ln}\:\mathrm{3}}}{\frac{\mathrm{ln}\:{b}}{\mathrm{ln}\:\mathrm{2}}}=\frac{\mathrm{ln}\:\mathrm{2}\:\mathrm{ln}\:{a}}{\mathrm{ln}\:\mathrm{3}\:\mathrm{ln}\:{b}} \\ $$ $$\frac{{m}}{{n}}=\frac{\frac{\mathrm{ln}\:\mathrm{3}\:\mathrm{ln}\:{a}}{\mathrm{ln}\:\mathrm{2}\:\mathrm{ln}\:{b}}}{\frac{\mathrm{ln}\:\mathrm{2}\:\mathrm{ln}\:{a}}{\mathrm{ln}\:\mathrm{3}\:\mathrm{ln}\:{b}}}=\frac{\left(\mathrm{ln}\:\mathrm{3}\right)^{\mathrm{2}} }{\left(\mathrm{ln}\:\mathrm{2}\right)^{\mathrm{2}} }=\left(\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{ln}\:\mathrm{2}}\right)^{\mathrm{2}} =\left(\mathrm{log}_{\mathrm{2}} \:\mathrm{3}\right)^{\mathrm{2}} \\ $$

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