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Question Number 172601 by Mikenice last updated on 29/Jun/22

Answered by Rasheed.Sindhi last updated on 30/Jun/22

$$\bullet {\log }_2\left(\frac{75}{16}\right)-{\log }_2{\left[\frac{{\left(\frac{25}{81}\right)}^{3/4}{\left(\frac{25}{81}\right)}^{1/4}}{{\left(\frac{125}{405}\right)}^{1/2}}\right]}^2$$$$+\frac{1}{3}({\log }_2{2}^{15}+{\log }_2{3}^{-15})$$$${\left[\frac{{\left(\frac{25}{81}\right)}^{3/4}{\left(\frac{25}{81}\right)}^{1/4}}{{\left(\frac{125}{405}\right)}^{1/2}}\right]}^2={\left[\frac{{\left(\frac{25}{81}\right)}^{\frac{3}{4}+\frac{1}{4}}}{{\left(\frac{125}{405}\right)}^{1/2}}\right]}^2$$$$=\frac{{\left(\frac{25}{81}\right)}^2}{{\left({\left(\frac{125}{405}\right)}^{1/2}\right)}^2}=\frac{5^4}{3^8}\times \frac{3^4\times 5}{5^3}=\frac{5^2}{3^4}$$$$\bullet {\log }_2\left(\frac{75}{16}\right)-{\log }_2\left(\frac{5^2}{3^4}\right)+\frac{1}{3}({\log }_2{2}^{15}+{\log }_2{3}^{-15})$$$$\bullet {\log }_2(\frac{75}{16}\mathbf{÷}\frac{5^2}{3^4})+\frac{1}{3}\left({\log }_2{2}^{15}\times 3^{-15}\right)$$$$\bullet {\log }_2\left(\frac{75}{16}\times \frac{3^4}{5^2}\right)+\frac{1}{3}({\log }_2\left(2^{15}\times 3^{-15}\right)$$$$\bullet {\log }_2\left(\frac{3^5}{2^4}\right)+{\log }_2\left(\frac{2^5}{3^5}\right)$$$$\bullet {\log }_2\left(\frac{3^5}{2^4}\times \frac{2^5}{3^5}\right)$$$$\bullet {\log }_2\left(2\right)=1$$

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