已知实数x .y.z.满足2^x=3^y=6^z 求证:xy+yz=xz
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已知实数x.y.z.满足2^x=3^y=6^z求证:xy+yz=xz已知实数x.y.z.满足2^x=3^y=6^z求证:xy+yz=xz已知实数x.y.z.满足2^x=3^y=6^z求证:xy+yz=
已知实数x .y.z.满足2^x=3^y=6^z 求证:xy+yz=xz
已知实数x .y.z.满足2^x=3^y=6^z 求证:xy+yz=xz
已知实数x .y.z.满足2^x=3^y=6^z 求证:xy+yz=xz
1/(lg2*lg3)-1/(lg3*lg6)=1/(lg2*lg6)
根据指数函数的特点,2^x、3^x、6^x的交点只有一个,就是x=0
所以x=y=z=0
自然有xy+yz=0=xz
得证
分别将上面的等式求以10为底的对数,xlg2=ylg3=zlg6,然后代入要证的等式的一边,进行化简就ok啦!
题目有问题
应该是xy-yz=xz
令2^x=3^y=6^z=k
则log2k=x
log3k=y
log6k=z
则 xy-yz=log2k*log3k-log3k*log6k=lgk/lg2*lgk/lg3-lgk/lg3*lgk/lg6
xz=lgk*lgk/(lg2*lg6)则只需要证明
1/(lg2*lg3)-1/(lg3*lg6)=1/(lg2*lg6),显然成立,证毕
证明有错误吧 应该证明 xy-yz=xz吧
设2^x=3^y=6^z =K
则x=Lnx/Ln2 y=Lnx/Ln3 z=Lnx/Ln6
整理后左边=右边 原题得证
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