Cho log 9 5 = a , log 4 7 =b , log 2 3 = c . Biết log 24 275 = mb + nac / pc + q
Phương pháp giải
Biến đổi log24175 theo a, b, c
Giải chi tiết
\({\rm{lo}}{{\rm{g}}_9}5 = a \Rightarrow \frac{1}{2}{\rm{lo}}{{\rm{g}}_3}5 = a \Rightarrow {\rm{lo}}{{\rm{g}}_3}5 = 2a \Rightarrow {\rm{lo}}{{\rm{g}}_5}3 = \frac{1}{{2a}}\)\({\rm{lo}}{{\rm{g}}_4}7 = b \Rightarrow \frac{1}{2}{\rm{lo}}{{\rm{g}}_2}7 = b \Rightarrow {\rm{lo}}{{\rm{g}}_2}7 = 2b \Rightarrow {\rm{lo}}{{\rm{g}}_7}2 = \frac{1}{{2b}}\)\({\rm{lo}}{{\rm{g}}_2}3 = c \Rightarrow {\rm{lo}}{{\rm{g}}_3}2 = \frac{1}{c}\)\( \Rightarrow {\rm{lo}}{{\rm{g}}_2}3 \cdot {\rm{lo}}{{\rm{g}}_7}2 = {\rm{lo}}{{\rm{g}}_7}3 = \frac{c}{{2b}}\)\( \Rightarrow {\rm{lo}}{{\rm{g}}_2}3 \cdot {\rm{lo}}{{\rm{g}}_3}5 = {\rm{lo}}{{\rm{g}}_2}5 = 2ac\)\({\rm{lo}}{{\rm{g}}_{24}}175 = 2{\rm{lo}}{{\rm{g}}_{24}}5 + {\rm{lo}}{{\rm{g}}_{24}}7 = \frac{2}{{{\rm{lo}}{{\rm{g}}_5}24}} + \frac{1}{{{\rm{lo}}{{\rm{g}}_7}24}}\)\( = \frac{2}{{3{\rm{lo}}{{\rm{g}}_5}2 + {\rm{lo}}{{\rm{g}}_5}3}} + \frac{1}{{3{\rm{lo}}{{\rm{g}}_7}2 + {\rm{lo}}{{\rm{g}}_7}3}}\)\( = \frac{2}{{3 \cdot \frac{1}{{2ac}} + \frac{1}{{2a}}}} + \frac{1}{{\frac{3}{{2b}} + \frac{c}{{2b}}}}\)\( = \frac{{4ac}}{{c + 3}} + \frac{{2b}}{{c + 3}} = \frac{{4ac + 2b}}{{c + 3}} = \frac{{mb + nac}}{{pc + q}} \Rightarrow n = 4;m = 2;p = 1,q = 3\)\(A = m + 2n + 3p + 4q = 2 + 8 + 3 + 12 = 25\)
Đáp án cần chọn là: B