设3的a次方=4的b次方=36,能否求出2/a+1/b的结果?若能求出,是多少?

题目

设3的a次方=4的b次方=36,能否求出2/a+1/b的结果?若能求出,是多少?

答案

3^a=36
lg3^a=lg6^2
alg3=2lg6=2lg2+2lg3
a=2lg2/lg3+2
4^b=36
lg4^b=lg6^2
lg2^2b=2lg6
2blg2=2lg2+2lg3
b=1+lg3/lg2
2/a+1/b
=(2b+a)/ab
=[2*(1+lg3/lg2)+lg2/lg3+2]/[(1+lg3/lg2)(lg2/lg3+2)]
=(2+2lg3/lg2+lg2/lg3+2)/(lg2/lg3+2+1+2lg3/lg2)
=(4+2lg3/lg2+lg2/lg3)/(3+2lg3/lg2+lg2/lg3)
=(4lg2lg3+3(lg3)^2+(lg2)^2)/(3lg2lg3+2(lg3)^2+(lg2)^2)
=(3lg3+lg2)(lg3+lg2)/(2lg3+lg2)(lg3+lg2)
=(3lg3+lg2)/(2lg3+lg2)