设x,y,z都是正数,且3^x=4^y=6^z,求证:1/z-1/x=1/2y∵x,y,z都是正数,且3^x=4^y=6^z ∴lg(3^x)=lg(4^y)=lg(6^z) 即: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
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![设x,y,z都是正数,且3^x=4^y=6^z,求证:1/z-1/x=1/2y∵x,y,z都是正数,且3^x=4^y=6^z ∴lg(3^x)=lg(4^y)=lg(6^z) 即: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](/uploads/image/z/14917942-46-2.jpg?t=%E8%AE%BEx%2Cy%2Cz%E9%83%BD%E6%98%AF%E6%AD%A3%E6%95%B0%2C%E4%B8%943%5Ex%3D4%5Ey%3D6%5Ez%2C%E6%B1%82%E8%AF%81%EF%BC%9A1%2Fz-1%2Fx%3D1%2F2y%E2%88%B5x%2Cy%2Cz%E9%83%BD%E6%98%AF%E6%AD%A3%E6%95%B0%2C%E4%B8%943%5Ex%3D4%5Ey%3D6%5Ez+%E2%88%B4lg%283%5Ex%29%3Dlg%284%5Ey%29%3Dlg%286%5Ez%29+%E5%8D%B3%EF%BC%9Axlg3%3Dylg4%3Dzlg6+%E8%AE%BExlg3%3Dylg4%3Dzlg6%3Dk+%E5%88%99x%3Dk%2Flg3%2Cy%3Dk%2Flg4%2Cz%3Dk%2Flg6+1%2Fz-1%2Fx+%3D1%2F%28k%2Flg6%29-1%2F%28k%2Flg3%29+%3Dlg6%2Fk-lg3%2Fk+%3D%28lg6-lg3%29%2Fk+%3Dlg2%2Fk)
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