a>b>0,ab=100,若函数f(x)=lg(x/a)乘以lg(x/b)的最小值为-0.25,求log(以a为第b)

lg(x/a)*lg(x/b)=(lgx-lga)(lgx-lgb)=(lgx)^2-lgx(lga+lgb)+lgalgb
移项化简，lg(b/a)=-1
log(以a为第b)=-1

f(x)=lg(x/a)*lg(x/b)
=(lgx-lga)(lgx-lgb)
=lg²x-(lga+lgb)lgx+lga*lgb
=lg²x-lg(ab)*lgx+lga*lgb
∵ab=100 ∴lg(ab)=2
∴f(x)=lg²x-2x+lga*lgb
=(lgx-1)²+lga*lgb-1
∵lgx∈R
∴当lgx=1时，
f(x)min=lga*lgb-1=-0.25
∴lga*lgb=3/4
∵ab=100，b=100/a
3/4=lga*lg(100/a)=lga*(2-lga)=2lga-lg²a，
4lg²a-8lga+3=0
解得：lga=1/2或：lga=3/2
所以lgb=3/2或lgb=1/2
因为a大于b大于0，所以 lga=3/2，lgb=1/2
∴根据换底公式有：log(a)b=lgb/lga=1/3

lg(x/a)*lg(x/b)=(lgx-lga)(lgx-lgb)
=(lgx)^2-lgx(lga+lgb)+lgalgb
min=[4lgalgb-(lga+lgb)^2]/4=-[(lga-lgb)^2]/4=-1/4
a>b>0 lga-lgb=1 lg(a/b)=1
所以,lg(b/a)=-1
log(以a为第b)=-1