1) Log5 1/7-log√5 25/√7 2) 8 в степени log4 3-1

Решите задачу:

$1)\; \; log_5\, \frac{1}{7}-log_{\sqrt5}\, \frac{25}{\sqrt7}=log_5\, 7^{-1}-(log_{5^{1/2}}\, 5^2-log_{5^{1/2}}7^{1/2})=\\\\\\\star \; \; log_{a^{k}}\, b^{n}=\frac{n}{k}\, log_{a}\, b\; \; \star \\\\\\=-log_57-\frac{2}{1/2}\, log_5\, 5+log_57=-log_57-4\underbrace {log_55}_{1}+log_57=-4$

$2)\; \; 8^{log_43-1}=(2^3)^{log_{2^2}3-1}=2^{3\cdot \frac{1}{2}\cdot log_2\, 3-3}=2^{log_2\, 3^{3/2}}\cdot 2^{-3}=\\\\=3^{\frac{3}{2}}\cdot \frac{1}{2^3}=\sqrt{3^3}\cdot \frac{1}{8}=\frac{3\sqrt3}{8}$