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已知：x,y,z均是正实数,且3^x=4^y=6^z.求证：1/z-1/x=1/2y
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已知：x,y,z均是正实数,且3^x=4^y=6^z.求证：1/z-1/x=1/2y
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答案和解析
设3^x=4^y=6^z=t,则
x=log3(t),y=1/2*log2(t),log3(2^z*3^z)=log3(t)→z=log3(t)/(log3(2)+1)
所以有左边=1/（log3(t)/(log3(2)+1)）-1/log3(t)
=(log3(2)+1)/log3(t)-1/log3(t)
=log3(2)/log3(t)
=logt(2)
同时有右边=1/（2*1/2*log2(t)）
=1/log2(t)
=log2(2)/log2(t)
=logt(2)
即左边=右边,所以1/z-1/x=1/2y得证
x=log3(t),y=1/2*log2(t),log3(2^z*3^z)=log3(t)→z=log3(t)/(log3(2)+1)
所以有左边=1/（log3(t)/(log3(2)+1)）-1/log3(t)
=(log3(2)+1)/log3(t)-1/log3(t)
=log3(2)/log3(t)
=logt(2)
同时有右边=1/（2*1/2*log2(t)）
=1/log2(t)
=log2(2)/log2(t)
=logt(2)
即左边=右边,所以1/z-1/x=1/2y得证
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