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

$${\log }_5\sqrt{27}\times {\log }_{9}125+{\log }_{16}12=...$$$$a.\frac{61}{36}$$$$b.\frac{9}{4}$$$$c.\frac{61}{20}$$$$d.\frac{41}{12}$$$$e.\frac{7}{2}$$

Answered by MJS last updated on 07/Aug/19

$${\log }_{b}a=\frac{\ln a}{\ln b}$$$$\frac{\ln \sqrt{27}}{\ln 5}\times \frac{\ln 125}{\ln 9}+\frac{\ln 12}{\ln 16}=\frac{\ln 3^{\frac{3}{2}}}{\ln 5}\times \frac{\ln 5^3}{\ln 3^2}+\frac{\ln 2^{2}3}{\ln 2^4}=$$$$=\frac{\frac{3}{2}\ln 3}{\ln 5}\times \frac{3\ln 5}{2\ln 3}+\frac{2\ln 2+\ln 3}{4\ln 2}=\frac{9}{4}+\frac{1}{2}+\frac{\ln 3}{4\ln 2}=$$$$=\frac{11}{4}+\frac{\ln 3}{4\ln 2}$$$$\mathrm{please}\mathrm{check}\mathrm{given}\mathrm{equation}$$

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