已知正整数x,y,z满足 3^x = 4^y = 6^z ,求证 1/z - 1/x = 1/2y.
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已知正整数x,y,z满足 3^x = 4^y = 6^z ,求证 1/z - 1/x = 1/2y.
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![已知正整数x,y,z满足 3^x = 4^y = 6^z ,求证 1/z - 1/x = 1/2y.](/uploads/image/z/19751335-7-5.jpg?t=%E5%B7%B2%E7%9F%A5%E6%AD%A3%E6%95%B4%E6%95%B0x%2Cy%2Cz%E6%BB%A1%E8%B6%B3+3%5Ex+%3D+4%5Ey+%3D+6%5Ez+%2C%E6%B1%82%E8%AF%81+1%2Fz+-+1%2Fx+%3D+1%2F2y.)
由题意3^x = 4^y = 6^z,
所以(同时取对数等式仍然成立)
得log(3^x) = log(4^y) = log(6^z)
化简得:xlog3=ylog4=zlog6
设辅助值结果为S;即:xlog3=ylog4=zlog6=S
所以得:
x=S / log3 y=S / log4 z=S / log6
所以:1/x=log3 / S 1/y=log4 / S 1/z=log6 / S
由:1/y=log4 / S得 1 / 2y=log4 / 2S
1/z - 1/x
=log6 /S – log3 /S=(log6 – log3) /S
=log(6/3) /S
= log2 /S
=2log2 / 2S
= log4 /2S
=1 / 2y
得证.
所以(同时取对数等式仍然成立)
得log(3^x) = log(4^y) = log(6^z)
化简得:xlog3=ylog4=zlog6
设辅助值结果为S;即:xlog3=ylog4=zlog6=S
所以得:
x=S / log3 y=S / log4 z=S / log6
所以:1/x=log3 / S 1/y=log4 / S 1/z=log6 / S
由:1/y=log4 / S得 1 / 2y=log4 / 2S
1/z - 1/x
=log6 /S – log3 /S=(log6 – log3) /S
=log(6/3) /S
= log2 /S
=2log2 / 2S
= log4 /2S
=1 / 2y
得证.
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