lg2/log5010+lg5/log510
=lg2/(lg10/lg50)+lg5/(lg10/lg5)
=lg2lg50+lg5lg5
=lg2(lg2+2lg5)+(lg5)^2
=(lg2)^2+2lg2lg5+(lg5)^2
=(lg2+lg5)^2
=(lg10)^2
=1