Question

设a，b，c均为实数，且满足a²+b²=c²求证：log2[1+（b+c/a）]+log2[1+（a

设a，b，c均为实数，且满足a²+b²=c²求证：log2[1+（b+c/a）]+log2[1+（a-c/b）]=1

Answer 1

已知：a²+b²=c²，那么：
log2 [1+（b+c/a）]+log2 [1+（a-c/b）]
=log2 [(a+b+c)/a ]+ log2 [(b+a-c)/b]
=log2 [(a+b+c)/a *(b+a-c)/b]
=log2 [(a+b)²-c²]/(ab)
=log2 [(a+b)²-a²-b²]/(ab)
=log2 (2ab)/(ab)
=log2 2
=1
所以等式得证。