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已知2^x=3^y=6^z≠1,求证1/x+1/y=1/z
题目详情
已知2^x=3^y=6^z≠1,求证1/x+1/y=1/z
▼优质解答
答案和解析
设2^x=3^y=6^z=r
→x=log(2)r,y=log(3)r,z=log(6)r
→1/x+1/y=1/log(2)r+1/log(3)r=log(3)2/log(3)r+log(3)3/log(3)r=log(3)6/log(3)r[期间要用换底公式]
对z:1/z=log(3)6/log(3)r
所以相等
→x=log(2)r,y=log(3)r,z=log(6)r
→1/x+1/y=1/log(2)r+1/log(3)r=log(3)2/log(3)r+log(3)3/log(3)r=log(3)6/log(3)r[期间要用换底公式]
对z:1/z=log(3)6/log(3)r
所以相等
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