Question

已知log5(3)=a,log5(2)=b,用a,b表示 1. log25(12) 2.log3(15) 3.log4(5/6) 4.log1.5(2.4)

求过程

Answer 1

∵log5(3) = lg3/lg5 = a ∴ lg3 = alg5
∵log5(2) = lg2/lg5 = b ∴ lg2 = blg5
∵log25(12) = lg12/lg25 = (lg3 + lg4)/(2lg5) = (lg3 + 2lg2)/lg5 = a + 2b
log3(15) = lg15/lg3 = (lg3+lg5)/lg3 = (a+1)/a
log4(5/6) = lg(5/6) / lg4 = (lg5 - lg6)/(2lg2) = (lg5 - lg2 + lg3)/(2lg2)
= (1 - b + a)/(2b)
log1.5(2.5) = lg2.5 / lg1.5 =lg(5/2) / lg(3/2) = (lg5-lg2) / (lg3-lg2) = (1-b)/(a-b)

追问

第四题的题目是log1.5(2.4)

追答

解：
∵log5(3) = lg3/lg5 = a ∴ lg3 = alg5
∵log5(2) = lg2/lg5 = b ∴ lg2 = blg5
∵log25(12) = lg12/lg25 = (lg3 + lg4)/(2lg5) = (lg3 + 2lg2)/lg5 = a + 2b
log3(15) = lg15/lg3 = (lg3+lg5)/lg3 = (a+1)/a
log4(5/6) = lg(5/6) / lg4 = (lg5 - lg6)/(2lg2) = (lg5 - lg2 + lg3)/(2lg2)
= (1 - b + a)/(2b)
log1.5(2.4) = lg(3/2) / lg(12/5) = (lg3 - lg2) / (lg3 + lg4 - lg5) = (lg3 - lg2)/(lg3 + 2lg2 - lg5)
= (a-b)/(a+2b-1)

用到的公式：
lga + lgb = lg(ab)
loga(b) = lgb/lga