a>0,b>0,log4 a=log6 a=log9 (a+b)=t 求a/b的值.

题目

a>0,b>0,log4 a=log6 a=log9 (a+b)=t 求a/b的值.

答案

log4 a = lga / lg4 = lga / (2lg2) = t -> lga = 2t * lg2
log6 b = lgb / lg6 = lgb / (lg2 + lg3) = t -> lgb = t * (lg2 + lg3)
log9 (a+b) = lg(a + b) / lg9 = lg(a + b) / (2lg3) = t -> lg(a + b) = 2t * lg3
-> lga + lg(a + b) = 2lgb
-> lg(a² + ab) = lgb²
-> a² + ab = b²
-> (a/b)² + a/b - 1 = 0
-> a/b = (√5 - 1) / 2 -(√5 + 1) / 2舍去