Question

Rút gọn biểu thức sau :

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

Answer 1

\(B=\left(\log b_a+\log_ba+2\right)\left(\log b_a-\log b_{ab}\right)-1=\left(\log b_a+\frac{1}{\log b_a}+2\right)\left(\log b_a.\log_ba-\left(\log_{ab}b.\log_ba\right)\right)-1\)

   \(=\frac{\log^2_ab+2\log_ab+1}{\log_ab}\left(1-\log_{ab}a\right)-1=\frac{\left(\log_ab+1\right)^2}{\log_ab}\left(1-\frac{1}{\log_aab}\right)-1\)

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