If-log-2-a-log-3-b-m-and-log-3-a-log-2-b-n-a-gt-1-and-b-gt-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-

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

If \frac{\log_{2} a}{\log_{3} b} = m and \frac{\log_{3} a}{\log_{2} b} = n

a > 1 and b > 1

then \frac{m}{n} = \dots

a . \log_{2} 3

b . \log_{3} 2

c . \log_{4} 9

d . {(\log_{2} 3)}^{2}

e . {(\log_{3} 2)}^{2}

Answered by MJS last updated on 07/Aug/19

m = \frac{\frac{\ln a}{\ln 2}}{\frac{\ln b}{\ln 3}} = \frac{\ln 3 \ln a}{\ln 2 \ln b}; n = \frac{\frac{\ln a}{\ln 3}}{\frac{\ln b}{\ln 2}} = \frac{\ln 2 \ln a}{\ln 3 \ln b}

\frac{m}{n} = \frac{\frac{\ln 3 \ln a}{\ln 2 \ln b}}{\frac{\ln 2 \ln a}{\ln 3 \ln b}} = \frac{{(\ln 3)}^{2}}{{(\ln 2)}^{2}} = {(\frac{\ln 3}{\ln 2})}^{2} = {(\log_{2} 3)}^{2}

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