Question

已知2的a次方×2的b次方=2的c次方×2的d次方=10,求证:(a-1)（d-1）=（b-1）（c-1）.

Answer 1

题目是不是打错了呢?2^a×5^b=2^c×5^d=10,求证（a-1）（d-1）=(b-1)(c-1)对么? 2^a×5^b=2^c×5^d=10 除以10 2^a×5^b/10=2^c×5^d/10=1 (2^a/2)×(5^b/5)=(2^c/2)×(5^d/5)=1 2^(a-1)×5^(b-1)=2^(c-1)×5^(d-1)=1 取对数 (a-1)lg2+(b-1)lg5=(c-1)lg2+(d-1)lg5=0 (a-1)lg2+(b-1)lg5=0 lg2/lg5=-(b-1)/(a-1) =(c-1)lg2+(d-1)lg5=0 lg2/lg5=-(d-1)/(c-1) 所以-(b-1)/(a-1)=-(d-1)/(c-1) (a-1)(d-1)=(b-1)(c-1)