Đề bài: Rút gọn biểu thức \(P = ({\log _a}b + {\log _b}a + 2)({\log _a}b – {\log _{ab}}b).{\log _b}a – 1.\)

\(\begin{array}{l} ({\log _a}b + {\log _b}a + 2)\left( {{{\log }_a}b – {{\log }_{ab}}b} \right){\log _b}a – 1\\ = ({\log _a}b + {\log _b}a + 2)\left( {{{\log }_a}b – \frac{{{{\log }_a}b}}{{{{\log }_a}(ab)}}} \right){\log _b}a – 1\\ = ({\log _a}b + \frac{1}{{{{\log }_a}b}} + 2)\left( {1 – \frac{1}{{1 + {{\log }_a}b}}} \right) – 1 \end{array}\)

Đặt \(t = {\log _a}b\)

\(\Rightarrow \left( {1 + \frac{1}{t} + 2} \right)\left( {1 – \frac{1}{{1 + t}}} \right) – 1 = \frac{{{t^2} + 2t + 1}}{t}.\frac{t}{{t + 1}} – 1 = t = {\log _a}b\)