Cho \({\log _a}b = 3,{\log _a}c = - 2\). Hãy tính \({\log _a}x\) với:
a) \(x = {a^3}{b^2}\sqrt c \)
b) \(x = \frac{{{a^4}\sqrt[3]{b}}}{{{c^3}}}\)
Áp dụng: \({\log _a}bc = {\log _a}b + {\log _a}c;{\log _a}\frac{b}{c} = {\log _a}b - {\log _a}c\); \({\log _a}{b^c} = c{\log _a}b\)
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a)
\(\begin{array}{l}{\log _a}\left( {{a^3}{b^2}\sqrt c } \right) = {\log _a}{a^3} + {\log _a}{b^2} + {\log _a}\sqrt c \\ = 3 + 2{\log _a}b + \frac{1}{2}{\log _a}c = 3 + 2.3 + \frac{1}{2}.\left( { - 2} \right) = 8\end{array}\)
b)
\(\begin{array}{l}{\log _a}\frac{{{a^4}\sqrt[3]{b}}}{{{c^3}}} = {\log _a}{a^4} + {\log _a}\sqrt[3]{b} - {\log _a}{c^3}\\ = 4 + \frac{1}{3}{\log _a}b - 3{\log _a}c = 4 + \frac{1}{3}.3 - 3.\left( { - 2} \right) = 11\end{array}\)