Bài 35: Trong mỗi trường hợp sau, hãy tính \({\log _a}x\) biết \({\log _a}b = 3,{\log _a}c = – 2\):
a) \(x = {a^3}{b^2}\sqrt c ;\) b) \(x = {{{a^4}\root 3 \of b } \over {{c^3}}}.\)
a) \({\log _a}x = {\log _a}\left( {{a^3}{b^2}\sqrt c } \right) \)
\(= 3 + 2{\log _a}b + {1 \over 2}{\log _a}c = 3 + 2.3 + {1 \over 2}\left( { – 2} \right) = 8\).
b) \({\log _a}x = {\log _a}\left( {{{{a^4}\root 3 \of b } \over {{c^3}}}} \right) \)
\(= 4 + {1 \over 3}{\log _a}b – 3{\log _a}c = 4 + {1 \over 3}.3 – 3\left( { – 2} \right) = 11\).
Bài 36: Trong mỗi trường hợp sau, hãy tìm x:
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a) \({\log _3}x = 4{\log _3}a + 7{\log _3}b\)
b) \({\log _5}x = 2{\log _5}a – 3{\log _5}b\)
a) \({\log _3}x = 4{\log _3}a + 7{\log _3}b = {\log _3}{a^4} + {\log _3}{b^7}\)
\(= {\log _3}\left( {{a^4}{b^7}} \right) \Rightarrow x = {a^4}{b^7}\)
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b) \({\log _5}x = 2{\log _5}a – 3{\log _5}b \)
\(= {\log _5}{{{a^2}} \over {{b^3}}} \Rightarrow x = {{{a^2}} \over {{b^3}}}.\)
Bài 37: Hãy biểu diễn các lôgarit sau qua \(\alpha \) và \(\beta \):
a) \({\log _{\sqrt 3 }}50\), nếu \({\log _3}15 = \alpha ,{\log _3}10 = \beta \);
b) \({\log _4}1250 = \alpha \), nếu \({\log _2}5 = \alpha \).
Áp dụng \({\log _{{a^\alpha }}}b = {1 \over \alpha }{\log _a}b\) \(\left( {a,b > 0,a \ne 1} \right)\)
a) \({\log _{\sqrt 3 }}50 = {\log _{{1 \over {{3^2}}}}}50 = 2{\log _3}50 = 2{\log _3}10 + 2{\log _3}5\)
\( = 2{\log _3}10 + 2{\log _3}{{15} \over 3} = 2{\log _3}10 + 2\left( {{{\log }_3}15 – 1} \right)\)
\( = 2\beta + 2\left( {\alpha – 1} \right) = 2\alpha + 2\beta – 2\)
b) \({\log _4}1250 = {1 \over 2}{\log _2}\left( {{5^4}.2} \right) = 2{\log _2}5 + {1 \over 2} = 2\alpha + {1 \over 2}.\)
