log(9)(a)=log(15)(b)=log(25)(a+2b),求b比a的值log(9)(a)=log(15)(b)=log(25)(a+2b),其中9,15和25是底数

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log(9)(a)=log(15)(b)=log(25)(a+2b),求b比a的值
log(9)(a)=log(15)(b)=log(25)(a+2b),其中9,15和25是底数

换底公式
lga/lg9=lgb/lg15=lg(a+2b)/lg25
令lga/lg9=lgb/lg15=lg(a+2b)/lg25=1/k
则lg9=k*lga
lg15=k*lgb
lg25=k*lg(a+2b)
lg9+lg25=lg9*25=lg225=lg15^2=2lg15
所以k*lga+k*lg(a+2b)=2k*lgb
lga+lg(a+2b)=2lgb
lg[a(a+2b)]=lgb^2
a^2+2ab=b^2
a^2+2ab-b^2=0
a=(-2b±√8b^2)/2=-b±√2*b
真数大于0
a>0,b>0
a/b>0
所以a/b=-1+√2
b/a=√2+1

log(9)(a)=1/log(a)(9)=1/2{log(a)(3)}
log(9)(a)=log(15)(b)=log(25)(a+2b)=k
9^k=a, 15^k=b
a+2b=25^k
9^k+2*15^k=25^k
(3/5)^k+2=(5/3)^k
(3/5)^k=y,(5/3)^k=1/y
y^2+2y+1=2
y=根号2-1 或y=1-根号2(<0)舍去
（3/5)k=根号2-1
k(lg3-lg5)=lg(根号2-1)
k=lg(根号2-1)/(lg3-lg5)....

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