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a) \(A=\log_{5^{-2}}5^{\frac{5}{4}}=-\frac{1}{2}.\frac{5}{4}.\log_55=-\frac{5}{8}\)
b) \(B=9^{\frac{1}{2}\log_22-2\log_{27}3}=3^{\log_32-\frac{3}{4}\log_33}=\frac{2}{3^{\frac{3}{4}}}=\frac{2}{3\sqrt[3]{3}}\)
c) \(C=\log_3\log_29=\log_3\log_22^3=\log_33=1\)
d) Ta có \(D=\log_{\frac{1}{3}}6^2-\log_{\frac{1}{3}}400^{\frac{1}{2}}+\log_{\frac{1}{3}}\left(\sqrt[3]{45}\right)\)
\(=\log_{\frac{1}{3}}36-\log_{\frac{1}{3}}20+\log_{\frac{1}{3}}45\)
\(=\log_{\frac{1}{3}}\frac{36.45}{20}=\log_{3^{-1}}81=-\log_33^4=-4\)
\(B=25^{\frac{1}{2}+\frac{1}{9}\log_{\frac{1}{2}}27+\log_{125}81}=\left(5^2\right)^{\frac{1}{2}+\frac{1}{9}\log_{5^{-1}}3^3+\log_{5^3}3^4}\)
\(=5^{1-\frac{2}{3}\log_53+\frac{8}{3}\log_53}=5^{1+2\log_53}=5.5^{\log_53^2}=5.9=45\)
\(D=\log_{5^{-1}}\left(5^2\right)-3\log_{3^2}\left(3^{-1}\right)+4.\log_{2^{\frac{3}{2}}}2^6=-2+\frac{3}{2}+16=\frac{31}{2}\)
Ta có :
\(\begin{cases}5>1;3>1\Rightarrow\log_53>0\\15>1;4>1\Rightarrow\log_{15}4>0\\0< \frac{1}{3}< 1;\frac{7}{2}>1\Rightarrow\log_{\frac{1}{3}}\frac{14}{5}< 0\\0< 0,3< 1;\frac{7}{2}>1\Rightarrow\log_{0,3}\frac{7}{2}< 0\end{cases}\)
\(\Rightarrow A=\frac{\log_53.\log_{15}4}{\log_{\frac{1}{3}}\frac{14}{5}\log_{0,3}\frac{7}{2}}>0\)
Theo công thức biến đổi có số ta có : \(\log_{a^n}x=\frac{\log_ax}{\log_aa^n}=\frac{1}{n}\log_ax\)
Từ đó ta có :
\(A=\frac{1}{\log_ax}+\frac{1}{\log_{a^2}x}+\frac{1}{\log_{a^3}x}+...+\frac{1}{\log_{a^n}x}\)
\(=\frac{1}{\log_ax}+\frac{2}{\log_ax}+\frac{4}{\log_ax}+...+\frac{n}{\log_ax}\)
\(=\frac{1+2+3+...+n}{\log_ax}=\frac{n\left(n+1\right)}{\log_ax}\)
Vậy \(A=\frac{1}{\log_ax}+\frac{1}{\log_{a^2}x}+\frac{1}{\log_{a^3}x}+...+\frac{1}{\log_{a^n}x}=\frac{n\left(n+1\right)}{\log_ax}\)
\(N=\log_{\frac{1}{3}}5\log_{25}\frac{1}{7}=\log_{3^{-1}}5\log_{5^5}3^{-3}=\left(-5\right)\left(-\frac{3}{2}\right).\log_35\log_53=\frac{15}{2}\)
Ta có : \(\log_{\frac{a}{b}}^2\frac{c}{b}=\log_{\frac{a}{b}}^2\frac{b}{c};\log_{\frac{b}{c}}^2\frac{a}{c}=\log_{\frac{b}{c}}^2\frac{c}{a};\log_{\frac{c}{a}}^2\frac{b}{a}=\log_{\frac{c}{a}}^2\frac{a}{b}\)
\(\Rightarrow\log_{\frac{a}{b}}^2\frac{c}{b}.\log_{\frac{b}{c}}^2\frac{a}{c}.\log_{\frac{c}{a}}^2\frac{b}{c}=\log_{\frac{a}{b}}^2\frac{c}{b}.\log^2_{\frac{b}{c}}\frac{c}{a}\log_{\frac{c}{a}}^2\frac{a}{b}=\left(\log_{\frac{a}{b}}\frac{c}{b}.\log_{\frac{b}{c}}\frac{c}{a}\log_{\frac{c}{a}}\frac{a}{b}\right)^2=1^2=1\)
\(\Rightarrow\) Trong 3 số không âm \(\log_{\frac{a}{b}}^2\frac{c}{b};\log^2_{\frac{b}{c}}\frac{c}{a};\log_{\frac{c}{a}}^2\frac{a}{b}\) luôn có ít nhất 1 số lớn hơn 1
\(\log_{\frac{1}{2}}\left(4^x+4\right)\ge\log_{\frac{1}{2}}\left(2^{x+1}-3\right)-\log_22^x\)
\(\Leftrightarrow\log_{\frac{1}{2}}\left(4^x+4\right)\ge\log_{\frac{1}{2}}\left(2^{x+1}-3\right)+\log_{\frac{1}{2}}2^x\)
\(\Leftrightarrow\log_{\frac{1}{2}}\left(4^x+4\right)\ge\log_{\frac{1}{2}}\left(2^{2x+1}-3^x\right)\)
\(\Leftrightarrow4^x+4\le2^{2x+1}-3.2^x\)
\(\Leftrightarrow4^x-3.2^x-4\ge0\)
\(\Leftrightarrow\begin{cases}2^x\le-1\left(L\right)\\2^x\ge4\end{cases}\)\(\Leftrightarrow x\ge2\)
Vậy bất phương trình có tập nghiệm \(S=\left(2;+\infty\right)\)
a. \(2^{2\log_25+\log_{\frac{1}{2}}9}\) và \(\frac{\sqrt{626}}{6}\)
Ta có : \(2^{2\log_25+\log_{\frac{1}{2}}9}=2^{\log_225-\log_29}=2^{\log_2\frac{25}{9}}=\frac{25}{9}=\frac{\sqrt{625}}{9}< \frac{\sqrt{626}}{6}\)
\(\Rightarrow2^{2\log_25+\log_{\frac{1}{2}}9}< \frac{\sqrt{626}}{6}\)
b. \(3^{\log_61,1}\) và \(7^{\log_60,99}\)
Ta có : \(\begin{cases}\log_61,1>0\Rightarrow3^{\log_61,1}>3^0=1\\\log_60,99< 0\Rightarrow7^{\log_60,99}< 7^0=1\end{cases}\)
\(\Rightarrow3^{\log_61,1}>7^{\log_60,99}\)
c. \(\log_{\frac{1}{3}}\frac{1}{80}\) và \(\log_{\frac{1}{2}}\frac{1}{15+\sqrt{2}}\)
Ta có : \(\begin{cases}\log_{\frac{1}{2}}\frac{1}{80}=\log_{3^{-1}}80^{-1}=\log_380< \log_381=4\\\log_{\frac{1}{2}}\frac{1}{15+\sqrt{2}}=\log_{2^{-1}}\left(15+\sqrt{2}\right)^{-1}=\log_2\left(15+\sqrt{2}\right)>\log_216=4\end{cases}\)
\(\Rightarrow\log_{\frac{1}{3}}\frac{1}{80}< \log_{\frac{1}{2}}\frac{1}{15+\sqrt{2}}\)
ta có
\(log_{\frac{1}{5}}125=-log_55^3=-3\)
\(log_{0.5}\frac{1}{2}=log_{0.5}0.5=1\)
\(log_{\frac{1}{4}}\frac{1}{64}=log_{\frac{1}{4}}\frac{1}{4^3}=3\)
\(log_{\frac{1}{6}}36=-log_636=-2\)