ta có \(\left(log^b_a+log^a_b+2\right)\left(log^b_a-log_{ab}^b\right).log_b^a-1=\left(log^b_a+log^a_b+2\right)\left(log^b_a.log_b^a-log_{ab}^b.log_b^a\right)-1=\left(log^b_a+log^a_b+2\right)\left(1-\frac{1}{log_b^{ba}}log_b^a\right)-1=\left(log^b_a+log^a_b+2\right)\left(1-\frac{1}{1+log^a_b}log^a_b\right)-1=\left(log^b_a+log^a_b+2\right)\frac{1}{1+log^a_b}-1=\left(log^a_b+\frac{1}{log^a_b}+2\right)\frac{1}{1+log^a_b}-1=\frac{\left(1+log^a_b\right)^2}{log^a_b}\frac{1}{1+log^a}-1=\frac{1+log^a_b}{log_b^a}-1=\frac{1}{log_b^a}\)

 ta có:

\(\left(log^b_a+\frac{1}{log^b_a}+2\right)\left(log^b_a-\frac{1}{log^{ab}_a}\right)log^a_b-1\)\(=\frac{\left(log^b_a+1\right)^2}{log^b_a}\left(log^b_a-\frac{1}{1+log^b_a}\right)log^a_b-1\)\(=\frac{\left(log^b_a+1\right)^2}{log^b_a}\left(1-\frac{log^a_b}{1+log^b_a}\right)-1\)\(==\frac{\left(log^b_a+1\right)^2}{log^b_a}\left(\frac{1}{1+log^b_a}\right)-1=\frac{1+log^b_a}{log^b_a}-1=\frac{1}{log^b_a}\)

Ta có \(A=\left(\log^3_ba+2\log^2_ba+\log_ba\right)\left(\log_ab-\log_{ab}b\right)-\log_ba\)

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

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

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

             \(=\log_ba+1-\log_ba=1\)

\(A=log_2\left(x^3-x\right)-log_2\left(x+1\right)-log_2\left(x-1\right)\)

\(=log_2\left(\dfrac{x^3-x}{x+1}\right)-log_2\left(x-1\right)\)

\(=log_2\left(\dfrac{x\left(x-1\right)\left(x+1\right)}{x+1}\right)-log_2\left(x-1\right)\)

\(=log_2\left(\dfrac{x\left(x-1\right)\left(x+1\right)}{\left(x+1\right)\left(x-1\right)}\right)=log_2x\)

a: \(log_49=\dfrac{log9}{log4}=\dfrac{log3^2}{log2^2}=\dfrac{2\cdot log3}{2\cdot log2}=\dfrac{log3}{log2}=\dfrac{b}{a}\)

b: \(log_612=\dfrac{log12}{log6}=\dfrac{log2^2+log3}{log2+log3}=\dfrac{2\cdot log2+log3}{log2+log3}\)

\(=\dfrac{2a+b}{a+b}\)

c: \(log_56=\dfrac{log6}{log5}=\dfrac{log\left(2\cdot3\right)}{log\left(\dfrac{10}{2}\right)}=\dfrac{log2+log3}{log10-log2}\)

\(=\dfrac{a+b}{1-a}\)

\(a,A=log_23\cdot log_34\cdot log_45\cdot log_56\cdot log_67\cdot log_78\\ =log_28\\ =log_22^3\\ =3\\ b,B=log_22\cdot log_24...log_22^n\\ =log_22\cdot log_22^2...log_22^n\\ =1\cdot2\cdot...\cdot n\\ =n!\)

\(A=\left(lna+log_{\alpha}e\right)^2+ln^2a-\log_a^2e\)

\(=ln^2a+\log_{\alpha}^2e+2\cdot lna\cdot\log_{\alpha}e+ln^2a-\log_{\alpha}^2e\)

\(=2\cdot\log_e^2\alpha+2\cdot\log_e\alpha\cdot\log_{\alpha}e\)

\(=2\cdot ln^2\alpha+2\)

 ta có:

\(log^{\left(2a^2\right)}_2+\left(log_2^a\right)a^{log_a^{\left(log^a_1+1\right)}}+\frac{1}{2}log^2_2a^4=log_2^2+log_2^{a^2}+log_2^a\left(log^a_2+1\right)+\frac{1}{2}log^2_2a^4\)

\(=1+2log^a_2+log^a_2\left(1+log^a_2\right)+2log^2a_2\)

\(=3log^2_2a+3log^a_2+1\)

\(P=loga^3+logb^2=log\left(a^3b^2\right)=log\left(100\right)=10\)

Lời giải:

Đặt \(\log_ab=x\Rightarrow \log_ba=\frac{1}{x}\)

a)

\(A=(x+\frac{1}{x}+2)(x-\frac{1}{x}).\frac{1}{x}\)

\(\Leftrightarrow A=(1+\frac{1}{x^2}+2x)(x-\frac{1}{x})=\left(1+\frac{1}{x}\right)^2(x-\frac{1}{x})\)

\(\Leftrightarrow A=(1+\log_ba)^2(\log_ab-\log_ba)\)

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b) Điều kiện: \(x>0\)

Có \(1=\log_{ab}b.\log_b(ab)=\log_{ab}b(\log_ba+\log_bb)=\log_{ab}b(\frac{1}{x}+1)\)

\(\Rightarrow \log_{ab}b=\frac{x}{x+1}\)

Như vậy:

\(B=\sqrt{x+\frac{1}{x}+2}(x-\frac{x}{x+1})\sqrt{x}\)

\(\Leftrightarrow B=\sqrt{x^2+1+2x}(x-\frac{x}{x+1})=|x+1|.\frac{x^2}{x+1}\)

\(=(x+1)\frac{x^2}{x+1}=x^2=\log_a^2b\) (do \(x>0)\)