Giải:

a) $\log_3 9^{1/2} = \frac{1}{2}\log_3 9 = \frac{1}{2}\cdot2 = 1.$
b) $\log_{\frac{1}{10}}\sqrt[3]{\frac{1}{25}} = \frac{1}{3}\log_{\frac{1}{10}}\frac{1}{25} = \frac{1}{3}\cdot2 = \frac{2}{3}.$
c) $\log_3 \sqrt[3]{9} = \frac{1}{3}\log_3 9 = \frac{1}{3}\cdot2 = \frac{2}{3}.$
d) $\sqrt{2+\sqrt{162-\sqrt{32}}} = \sqrt{2+\sqrt{162-4\sqrt{2}}}$ (rút gọn tuỳ theo yêu cầu).
e) $(\sqrt[3]{5})^4 \sqrt[3]{8^5} = 5^{4/3} \cdot 8^{5/6} = 5^{4/3} \cdot 2^{5/2}.$

Giải:

a) $\log_3 45 + \log_3 \frac{1}{5} = \log_3 (45\cdot\frac{1}{5}) = \log_3 9 = 2.$
b) $\log_3 48 - \log_3 3 = \log_3 \frac{48}{3} = \log_3 16 = 4\log_3 2.$
c) $\log_3 16 + 2\log_3 \sqrt{6} = \log_3 16 + \log_3 6 = \log_3 96.$
d) $\log_{\frac{1}{3}} 9 + \log_{\frac{1}{3}} 27 - \log_{\frac{1}{3}} 7 = -2 + (-3) - (-\log_3 7) = -5 + \log_3 7.$

Giải:

a) $\log_3 \frac{1}{27} = \log_3 3^{-3} = -3.$
b) $\log_9 \frac{1}{16} = \log_9 2^{-4} = -4\log_9 2.$
c) $\log_{27} 10^5 \log_5 8 = \frac{5\log 10}{\log 27}\cdot\frac{\log 8}{\log 5} = \frac{5\cdot3}{3\log 3} \cdot \frac{3\log 2}{\log 5}.$

Giải:

Gọi $\log 2 = \frac{a}{2}, \log 3 = b.$
a) $\log 18 = \log(2\cdot3^2) = \log 2 + 2\log 3 = \frac{a}{2} + 2b.$
b) $\log 12 = \log(2^2\cdot3) = 2\log 2 + \log 3 = a + b.$
c) $\log 75 = \log(3\cdot5^2) = \log 3 + 2(\log 10 - \log 2) = b + 2(1 - \frac{a}{2}) = 2 - a + b.$