Chương 2: HÀM SỐ LŨY THỪA. HÀM SỐ MŨ VÀ HÀM SỐ LÔGARIT

Ta có :

\(M=\frac{7\ln\left(\sqrt{2}+1\right)^2-64\ln\left(\sqrt{2}+1\right)-50\ln\left(\sqrt{2}+1\right)^{-1}+2}{-3lg5-lg\left(10^{-1}.2^3\right)+6lg\left(10^{-\frac{1}{3}}.2^{\frac{2}{3}}\right)+4lg\left(10.5\right)}\)

    \(=\frac{2}{lg5+1-3lg2-2+4lg2+4}=\frac{1}{2}\)

Đặt \(A=\sqrt[4]{2+\sqrt{5}+2\sqrt{2+\sqrt{5}}};B=\sqrt[4]{2+\sqrt{5}-2\sqrt{2+\sqrt{5}}}\)

\(\Rightarrow N=A+B\)

Ta có \(AB=\sqrt[4]{\left(2+\sqrt{5}\right)^2-4\left(\sqrt{2+\sqrt{5}}\right)}=1\)

và \(A^4+B^4=4+2\sqrt{5}\)

Suy ra \(A^4+B^4=2A^2B^2=6+2\sqrt{5}=\left(\sqrt{5}+1\right)^2\)

          \(\Leftrightarrow A^2+B^2=\sqrt{5}+1\)

Tức là :

\(A^2+B^2+2AB=\sqrt{5}+3=\left(\frac{\sqrt{5}+1}{\sqrt{2}}\right)^2\)

\(\Leftrightarrow A+B=\frac{\sqrt{5}+1}{\sqrt{2}}\)

Vậy \(N=\frac{\sqrt{5}+1}{\sqrt{2}}\)

\(R=\log_22x^2+\left(\log_2x\right)x^{\log_x\left(\log_2x+1\right)}+\frac{1}{2}\log^2_4x^4\)

    \(=1+2\log_2x+\left(\log_2x\right)\left(\log_2x+1\right)+2\log^2_2x\)

    \(=3\log^2_2x+3\log_2x+1\)

\(=\left(\log_ab+\log_ba+2\right)\left(1-\log_{ab}a\right)-1\)

\(=\left(\log_ab+\log_ba+2\right)\left(1-\frac{1}{1+\log_ab}\right)-1\)

\(=\frac{1}{1+\log_ab}\left(\log_ab+\log_ba+2\right)-1\)

\(=\frac{1}{1+\log_ab}\left[\left(\log_ab+\log_ba+2\right)-1-\log_ab\right]\)

\(=\frac{1}{1+\log_ab}\left(\log_ab+\log^2_ba\right)=\log_ab\)

Ta có \(a=\frac{1}{2}\log_711;b=\log_27\)

Mặt khác  : \(\log_{\sqrt[3]{7}}\frac{121}{8}=3\log_7\frac{11^2}{2^3}=3\left(2\log_711-3\log_72\right)=6\log_711-\frac{9}{\log_27}=12a-\frac{9}{b}\)

Vậy \(\log_{\sqrt[3]{7}}\frac{121}{8}=12a-\frac{9}{b}\)

Điều kiện : \(\begin{cases}\ln\frac{1}{x-1}\ge0\\x-1>0\end{cases}\)\(\Leftrightarrow\begin{cases}\frac{1}{x-1}\ge1\\x>1\end{cases}\) \(\Leftrightarrow1< x\le2\)

Vậy tập xác định : \(D=\) (1;2]

Ta có : \(y'=\cos x.e^{\sin x}\Rightarrow y"=-\sin x.e^{\sin x}+\cos^2x.e^{\sin x}\)

       \(\Rightarrow y"=-\sin x.y+\cos x.y'\Rightarrow y'\cos x-y.\sin x-y"=0\)

=> Điều phải chứng minh

\(y'=\frac{3\cos3x}{10\sqrt[10]{\sin^93x}}\)

\(y'=\frac{1-\ln x-\left(1-\ln x-1\right)}{x^2\left(1-\ln x\right)^2}=\frac{1}{x^2\left(1-\ln x\right)^2}\)

\(A=\log_{\frac{\sqrt{b}}{a}}\frac{\sqrt[3]{b}}{\sqrt{a}}=\log_{\frac{\sqrt{b}}{a}}b^{\frac{1}{3}}-\log_{\frac{\sqrt{b}}{a}}a^{\frac{1}{3}}=\frac{1}{3\log_b\frac{\sqrt{b}}{a}}-\frac{1}{2\log_a\frac{\sqrt{b}}{a}}\)

    \(=\frac{1}{3\left(\frac{1}{2}-\log_ba\right)}-\frac{1}{2\left(\frac{1}{2}\log_ab-1\right)}\)

    \(=\frac{1}{3\left(\frac{1}{2}-\log_ba\right)}-\frac{1}{\log_ab-2}=\frac{a\log_ab}{3\left(\log_ab-2\right)}-\frac{1}{\log_ab-2}\)

   \(=\frac{2\sqrt{3}-3}{3\left(\sqrt{3}-2\right)}=-\frac{\sqrt{3}}{3}\)