log-5-27-log-9-125-log-16-12-a-61-36-b-9-4-c-61-20-d-41-12-e-7-2-

Question Number 65970 by gunawan last updated on 07/Aug/19

\log_{5} \sqrt{27} \times \log_{9} 125 + \log_{16} 12 = \dots

a . \frac{61}{36}

b . \frac{9}{4}

c . \frac{61}{20}

d . \frac{41}{12}

e . \frac{7}{2}

Answered by MJS last updated on 07/Aug/19

\log_{b} a = \frac{\ln a}{\ln b}

\frac{\ln \sqrt{27}}{\ln 5} \times \frac{\ln 125}{\ln 9} + \frac{\ln 12}{\ln 16} = \frac{\ln 3^{\frac{3}{2}}}{\ln 5} \times \frac{\ln 5^{3}}{\ln 3^{2}} + \frac{\ln 2^{2} 3}{\ln 2^{4}} =

= \frac{\frac{3}{2} \ln 3}{\ln 5} \times \frac{3 \ln 5}{2 \ln 3} + \frac{2 \ln 2 + \ln 3}{4 \ln 2} = \frac{9}{4} + \frac{1}{2} + \frac{\ln 3}{4 \ln 2} =

= \frac{11}{4} + \frac{\ln 3}{4 \ln 2}

please check given equation

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