Question

若a,b均为不等于1的正数,且logba(以b为底a的对数)+ logab(以a为底b的对数)=5/2,求(a^3+b^

Answer 1

令logb(a)=t,则loga(b)=1/t

t+1/t=5/2

整理,得

2t^2-5t+2=0

(t-2)(2t-1)=0

t=2或t=1/2

由所求式对称性,不妨设a=b^2

(a^3+b^3)/(ab+a^2b^2)

=b^3(b^3+1)/[b^3(1+b^3)]

=1