Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 314

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$$1)\frac{{\log }_{5}2}{{\log }_{5}6}+\frac{{\log }_{4}3}{{\log }_{4}6}=$$

$$={\log }_{6}2+{\log }_{6}3={\log }_6(2\bullet 3)=$$

$$={\log }_{6}6=1$$

$$2)({\log }_{7}2+\frac{1}{{\log }_{5}7})\bullet \lg 7=$$

$$=({\log }_{7}2+\frac{{\log }_{5}5}{{\log }_{5}7})\bullet \lg 7=$$

$$=({\log }_{7}2+{\log }_{7}5)\bullet \frac{{\log }_{7}7}{{\log }_{7}10}=$$

$$={\log }_7(2\bullet 5)\bullet \frac{1}{{\log }_{7}10}=$$

$$=\frac{{\log }_{7}10}{{\log }_{7}10}=1$$

$$3)\frac{2{\log }_{2}3}{{\log }_{4}9}=\frac{2{\log }_{2}3}{{\log }_{2^2}9}=$$

$$=\frac{2{\log }_{2}3}{\frac{1}{2}{\log }_{2}9}=4{\log }_{9}3=$$

$$=4{\log }_9{9}^{\frac{1}{2}}=4\bullet \frac{1}{2}=2$$