Cho a = log 2 3 ; b = log 3 5 ; c = log 7 2. Tính giá trị của log 140 63 theo a , b , c .

2. (0,5 điểm)

Ta có \[{\log _{140}}63 = \frac{{{{\log }_2}63}}{{{{\log }_2}140}} = \frac{{{{\log }_2}\left( {{3^2} \cdot 7} \right)}}{{{{\log }_2}\left( {{2^2} \cdot 5 \cdot 7} \right)}} = \frac{{2{{\log }_2}3 + {{\log }_2}7}}{{2 + {{\log }_2}5 + {{\log }_2}7}}\]

                     \[ = \frac{{2{{\log }_2}3 + \frac{1}{{{{\log }_7}2}}}}{{2 + {{\log }_2}3 \cdot {{\log }_3}5 + \frac{1}{{{{\log }_7}2}}}} = \frac{{2a + \frac{1}{c}}}{{2 + ab + \frac{1}{c}}}\]

                     \[ = \frac{{1 + 2ac}}{{1 + 2c + abc}}.\]