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已知x、y、z为正数且3^x=4^y=6^z,求证：1/2y=1/z-1/x
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已知x、y、z为正数且3^x=4^y=6^z,求证：1/2y=1/z-1/x
▼优质解答
答案和解析
xlg3=ylg4=zlg6
设xlg3=ylg4=zlg6=k
则x=k/lg3,y=k/lg4,z=k/lg6
1/z-1/x =1/(k/lg6)-1/(k/lg3) =lg6/k-lg3/k =(lg6-lg3)/k =lg2/k
1/2y =1/(2k/lg4) =lg4/(2k) =2lg2/(2k) =lg2/k
∴1/z-1/x=1/2y
设xlg3=ylg4=zlg6=k
则x=k/lg3,y=k/lg4,z=k/lg6
1/z-1/x =1/(k/lg6)-1/(k/lg3) =lg6/k-lg3/k =(lg6-lg3)/k =lg2/k
1/2y =1/(2k/lg4) =lg4/(2k) =2lg2/(2k) =lg2/k
∴1/z-1/x=1/2y
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