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Chọn 2 làm cơ số, ta có :
\(A=\log_616=\frac{\log_216}{\log_26}=\frac{4}{1=\log_23}\)
Mặt khác :
\(x=\log_{12}27=\frac{\log_227}{\log_212}=\frac{3\log_23}{2+\log_23}\)
Do đó : \(\log_23=\frac{2x}{3-x}\) suy ra \(A=\frac{4\left(3-x\right)}{3+x}\)
b) Ta có :
\(B=\frac{lg30}{lg125}=\frac{lg10+lg3}{3lg\frac{10}{2}}=\frac{1+lg3}{3\left(1-lg2\right)}=\frac{1+a}{3\left(1-b\right)}\)
c) Ta có :
\(C=\log_65+\log_67=\frac{1}{\frac{1}{\log_25}+\frac{1}{\log_35}}+\frac{1}{\frac{1}{\log_27}+\frac{1}{\log_37}}\)
Ta tính \(\log_25,\log_35,\log_27,\log_37\) theo a, b, c .
Từ : \(a=\log_{27}5=\log_{3^3}5=\frac{1}{3}\log_35\)
Suy ra \(\log_35=3a\) do đó :
\(\log_25=\log_23.\log35=3ac\)
Mặt khác : \(b=\log_87=\log_{2^3}7=\frac{1}{3}\log_27\) nên \(\log_27=3b\)
Do đó : \(\log_37=\frac{\log_27}{\log_23}=\frac{3b}{c}\)
Vậy : \(C=\frac{1}{\frac{1}{3ac}+\frac{1}{3a}}+\frac{1}{\frac{1}{3b}+\frac{c}{3b}}=\frac{3\left(ac+b\right)}{1+c}\)
d) Điều kiện : \(a>0;a\ne0;b>0\)
Từ giả thiết \(\log_ab=\sqrt{3}\) suy ra \(b=a^{\sqrt{3}}\). Do đó :
\(\frac{\sqrt{b}}{a}=a^{\frac{\sqrt{3}}{2}-1};\frac{\sqrt[3]{b}}{\sqrt{a}}=a^{\frac{\sqrt{3}}{3}-\frac{1}{2}}=a^{\frac{\sqrt{3}}{3}\left(\frac{\sqrt{3}}{2}-1\right)}\)
Từ đó ta tính được :
\(A=\log_{a^{\alpha}}a^{\frac{-\sqrt{3}}{3}\alpha}=\log_{a^{\alpha}}\left(a^{\alpha}\right)^{\frac{-\sqrt{3}}{3}}=\frac{-\sqrt{3}}{3}\) với \(\alpha=\frac{\sqrt{3}}{2}-1\)
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 \(D=\log_{\frac{1}{3}}6^2-\log_{\frac{1}{3}}400^{\frac{1}{2}}+\log_{\frac{1}{3}}\left(\sqrt[3]{45}\right)\)
\(=\log_{\frac{1}{3}}36-\log_{\frac{1}{3}}20+\log_{\frac{1}{3}}45\)
\(=\log_{\frac{1}{3}}\frac{36.45}{20}=\log_{3^{-1}}81=-\log_33^4=-4\)
Cho \(\log_ab=3;\log_ac=-2\)
1. Với \(x=a^3b^2\sqrt{c}\Rightarrow\log_ax=\log_a\left(a^3b^2\sqrt{c}\right)=\log_aa^3+\log_ab^2+\log_ac^{\frac{1}{2}}\)
\(=3+2.3+\frac{1}{2}\left(-2\right)=8\)
2. Với \(x=\frac{a^4\sqrt[3]{b}}{c^3}\) \(\Rightarrow\log_a\frac{a^4\sqrt[3]{b}}{c^2}=\log_aa^4+\log_ab^{\frac{1}{3}}+\log_ac^3\)
\(=4+\frac{1}{3}\log_ab+3\log_ac=4+\frac{1}{3}.3+3\left(-2\right)=-1\)
3. Với \(x=\log_a\frac{a^2\sqrt[3]{b}c}{\sqrt[3]{a\sqrt{c}}b^3}\Rightarrow\log_a\frac{a^2b^{\frac{1}{3}}c}{a^{\frac{1}{3}}b^3c^{\frac{1}{6}}}=\log_a\frac{a^{\frac{5}{3}}c^{\frac{5}{6}}}{b^{\frac{8}{3}}}=\log_aa^{\frac{5}{3}}-\log_ab^{\frac{8}{3}}+\log_ac^{\frac{3}{2}}\)
\(=\frac{5}{3}-\frac{8}{3}\log_ab+\frac{5}{6}\log_ac=\frac{5}{3}-\frac{8}{3}3+\frac{5}{6}\left(-2\right)=-8\)
a) \(A=\frac{a^{\frac{5}{2}}\left(a^{\frac{1}{2}}-a^{\frac{-3}{2}}\right)}{a^{\frac{1}{2}}\left(a^{\frac{-1}{2}}-a^{\frac{3}{2}}\right)}=\frac{a^3-a}{1-a^2}=-a\)
Do đó : \(A=-\left(\pi-3\sqrt{2}\right)=3\sqrt{2}-\pi\)
b) Rút gọn B ta có :
\(B=\left(a^{\frac{1}{3}}+b^{\frac{1}{3}}\right)\left[\left(a^{\frac{1}{3}}\right)^2+\left(b^{\frac{1}{3}}\right)^2\right]=\left(a^{\frac{1}{3}}\right)^3+\left(b^{\frac{1}{3}}\right)^3=a+b\)
Do đó :
\(B=\left(7-\sqrt{2}\right)+\left(\sqrt{2}+3\right)=10\)
Ta có \(a=\log_{\sqrt{2}}\left(\frac{1}{\sqrt[3]{5}}\right)=\log_{2^{\frac{1}{2}}}5^{-\frac{1}{3}}=-\frac{2}{3}\log_25\)
\(\Rightarrow\log_25=-\frac{3a}{2}\)
\(\Rightarrow C=\log40=\frac{\log_240}{\log_210}=\frac{\log_2\left(2^3.5\right)}{\log_2\left(2.5\right)}=\frac{3+\log_25}{1+\log_25}=\frac{6-3a}{2-3a}\)
\(E=16\left[\log_{3^{-2}}3^{\frac{3}{2}}\right]^2+23\log_{2^{\frac{9}{2}}}2^{\frac{5}{2}}-12\log_55^{-3}=16\left(-\frac{3}{4}\right)^2+9\frac{5}{9}-12\left(-3\right)=50\)
a) Áp dụng bất đẳng thức Cauchy cho các số dương, ta có :
\(\log_23+\log_32>2\sqrt{\log_23.\log_32}=2\sqrt{1}=2\)
Không xảy ra dấu "=" vì \(\log_23\ne\log_32\)
Mặt khác, ta lại có :
\(\log_23+\log_32<\frac{5}{2}\Leftrightarrow\log_23+\frac{1}{\log_23}-\frac{5}{2}<0\)
\(\Leftrightarrow2\log^2_23-5\log_23+2<0\)
\(\Leftrightarrow\left(\log_23-1\right)\left(\log_23-2\right)<0\) (*)
Hơn nữa, \(2\log_23>2\log_22>1\) nên \(2\log_23-1>0\)
Mà \(\log_23<\log_24=2\Rightarrow\log_23-2<0\)
Từ đó suy ra (*) luôn đúng. Vậy \(2<\log_23+\log_32<\frac{5}{2}\)
b) Vì \(a,b\ge1\) nên \(\ln a,\ln b,\ln\frac{a+b}{2}\) không âm.
Áp dụng bất đẳng thức Cauchy ta có
\(\ln a+\ln b\ge2\sqrt{\ln a.\ln b}\)
Suy ra
\(2\left(\ln a+\ln b\right)\ge\ln a+\ln b+2\sqrt{\ln a\ln b}=\left(\sqrt{\ln a}+\sqrt{\ln b}\right)^2\)
Mặt khác :
\(\frac{a+b}{2}\ge\sqrt{ab}\Rightarrow\ln\frac{a+b}{2}\ge\frac{1}{2}\left(\ln a+\ln b\right)\)
Từ đó ta thu được :
\(\ln\frac{a+b}{2}\ge\frac{1}{4}\left(\sqrt{\ln a}+\sqrt{\ln b}\right)^2\)
hay \(\frac{\sqrt{\ln a}+\sqrt{\ln b}}{2}\le\sqrt{\ln\frac{a+b}{2}}\)
c) Ta chứng minh bài toán tổng quát :
\(\log_n\left(n+1\right)>\log_{n+1}\left(n+2\right)\) với mọi n >1
Thật vậy,
\(\left(n+1\right)^2=n\left(n+2\right)+1>n\left(n+2\right)>1\)
suy ra :
\(\log_{\left(n+1\right)^2}n\left(n+2\right)<1\Leftrightarrow\frac{1}{2}\log_{n+1}n\left(n+2\right)<1\)
\(\Leftrightarrow\log_{n+1}n+\log_{\left(n+1\right)}n\left(n+2\right)<2\)
Áp dụng bất đẳng thức Cauchy ta có :
\(2>\log_{\left(n+1\right)}n+\log_{\left(n+1\right)}n\left(n+2\right)>2\sqrt{\log_{\left(n+1\right)}n.\log_{\left(n+1\right)}n\left(n+2\right)}\)
Do đó ta có :
\(1>\log_{\left(n+1\right)}n.\log_{\left(n+1\right)}n\left(n+2\right)\) và \(\log_n\left(n+1>\right)\log_{\left(n+1\right)}\left(n+2\right)\) với mọi n>1
a. \(0,7^{\frac{\sqrt{5}}{2}}\) và \(0,7^{\frac{1}{3}}\).
Ta có : \(\begin{cases}\left(\frac{\sqrt{5}}{6}\right)^2=\frac{5}{36}>\frac{4}{36}=\left(\frac{1}{3}\right)^2\Rightarrow\frac{\sqrt{5}}{6}>\frac{1}{3}\\0< 0,7< 1\end{cases}\)
\(\Rightarrow0,7^{\frac{\sqrt{5}}{6}}< 0,7^{\frac{1}{3}}\)
b. \(2^{\sqrt{3}}\) và \(3^{\sqrt{2}}\)
Ta có : \(\begin{cases}\left(2^{\sqrt{3}}\right)^{\sqrt{3}}=2^3=8\\\left(3^{\sqrt{2}}\right)^{\sqrt{3}}=3^{\sqrt{6}}>3^2=9\end{cases}\)
\(\Rightarrow\left(2^{\sqrt{3}}\right)^{\sqrt{3}}< \left(3^{\sqrt{2}}\right)^{\sqrt{3}}\)
\(\Rightarrow2^{\sqrt{3}}< 3^{\sqrt{2}}\)
c. \(\log_{0.4}\sqrt{2}\) và \(\log_{0,2}0,34\)
Ta có : \(\begin{cases}0< 0,4< 1;\sqrt{2}>1\Rightarrow\log_{0,4}\sqrt{2}< 0\\0< 0,2< 1;0< 1< 0,34\Rightarrow\log_{0,2}0,3>0\end{cases}\)
\(\Rightarrow\log_{0,4}\sqrt{2}< \log_{0,2}0,34\)
\(A=\log_{\frac{\sqrt{b}}{a}}\frac{\sqrt[3]{b}}{\sqrt{a}}=\log_{\frac{\sqrt{b}}{a}}b^{\frac{1}{3}}-\log_{\frac{\sqrt{b}}{a}}a^{\frac{1}{3}}=\frac{1}{3\log_b\frac{\sqrt{b}}{a}}-\frac{1}{2\log_a\frac{\sqrt{b}}{a}}\)
\(=\frac{1}{3\left(\frac{1}{2}-\log_ba\right)}-\frac{1}{2\left(\frac{1}{2}\log_ab-1\right)}\)
\(=\frac{1}{3\left(\frac{1}{2}-\log_ba\right)}-\frac{1}{\log_ab-2}=\frac{a\log_ab}{3\left(\log_ab-2\right)}-\frac{1}{\log_ab-2}\)
\(=\frac{2\sqrt{3}-3}{3\left(\sqrt{3}-2\right)}=-\frac{\sqrt{3}}{3}\)