abc都是正数,3的a次方=4的b次方=6的c次方.求证c分之2等于a分之2加b分之1
问题描述：
abc都是正数,3的a次方=4的b次方=6的c次方.求证c分之2等于a分之2加b分之1
答
aln3=bln4=cln6=t
1/c=ln6/t
1/a=ln3/t
1/b=ln4/t
2/a+1/b=2ln3/t+2ln2/t=2ln6/t=2/c
答
3^a=4^b=6^c
a*lg3=b*lg4=c*lg6
a*lg3=2b*lg2=c*(lg2+lg3)
a=c*(lg2+lg3)/lg3
b=c*(lg2+lg3)/(2*lg2)
2/a+1/b
=2/{c*(lg2+lg3)/lg3}+1/[c*(lg2+lg3)/(2*lg2)]
=2*lg3/[c*(lg2+lg3)]+2*lg2/[c*(lg2+lg3)]
=2*(lg2+lg3)/[c*(lg2+lg3)]
=2/c
答
3^a=4^b=6^calg3=blg4=clg6alg3=2blg2=c(lg2+lg3)(a-c)lg3=clg2 1)clg3=(2b-c)lg2 2)1)÷2）,得(a-c)/c=c/(2b-c)c^2=(a-c)(2b-c)=2ab-2bc-ac+c^2所以2ab-2bc-ac=0两边同时除以abc,得2/c-2/a-1/b=0即2/c=2/a+1/b...