2^x=5^y=10^z(xyz不等于0）求证xy=yz+xzrt

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2^x=5^y=10^z(xyz不等于0）求证xy=yz+xzrt
2^x=5^y=10^z(xyz不等于0）求证xy=yz+xz
rt

2^x=5^y=10^z(xyz不等于0）求证xy=yz+xzrt
2^x=10^z
(2^x)^y=(10^z)^y
2^xy=10^yz
5^y=10^z
(5^y)^x=(10^z)^x
5^xy=10^xz
相乘
2^xy*5^xy=10^yz*10^xz
(2*5)^xy=10^yz*10^xz
10^xy=10^(yz+xz)
xy=yz+xz

2^x=5^y=10^z=k
x=log2 k
y=log5 k
z=lgk
1/x+1/y=logk 2+logk 5=logk (2*5)=logk 10
1/z=logk 10
所以，1/x+1/y=1/z
xy=yz+xz

把上式2边取10的对数即X*lg2=Y*lg5=Z 再把X*lg2=Y*lg5式子2边加上Xlg5得 X*lg2+X*lg5=X*lg5+Y*lg5 X=(X+Y)*lg5得lg5=X/(X+Y)把它代入Y*lg5=Z中得X*Y/(X+Y)=Z整理下得到 XY=YZ+XZ

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