y=a^x(a>1)的定义域与值域均为【m,n】，求a的取值范围

y=a^x是单调增的, ∴a^m=m,a^n=n

令f(x)=a^x-x, 由题意,f(x)=0有两个不等实根m,n

f'(x)=lna×a^x-1

令f'(x)>=0, 则a^x>=1/lna=loga e, ∴x>=loga (loga e); 令f'(x)<=0, 则x<=loga (loga e)

∴f(x)在(-∞, loga (loga e)]上单调减,在[loga (loga e),+∞)上单调增

要使f(x)=0有两个不等实根, 则f(x)在x=loga (loga e)处小于0

f(loga (loga e))=a^(loga (loga e))-loga (loga e)=loga e-loga (loga e)<0

∴loga e<loga (loga e), e<loga e

∴a^e<a^(loga e)=e, ∴(a^e)^(1/e)<e^(1/e), 即a<e^(1/e)

综上,1<a<e^(1/e), 即取值范围为(1,e^(1/e))