推 tzhau :感謝 02/01 23:51
※ 引述《tzhau (生命中無法承受之輕)》之銘言:
: 1.用1、5、8、9這四個數字構成a^[b^(c^d)]之形式,且a、b、c、d互異,求其最大值
: 2.三個正整數a、b、c滿足條件:
: (1)a<b<c<30 (2)以某一正整數為底,a(2b-a)與c^2+60b-11a的對數分別為9和11
: 試求a,b,c之值
: 3.已知 x(y+z-x)/log x = y(z+x-y)/log y = z(x+y-z)/log z 且 xyz=1,求證:
: (x^y)(y^x)=(y^z)(z^y)=(z^x)(x^z)
x(y+z-x)/log x = y(z+x-y)/log y = z(x+y-z)/log z = k
則 x(y+z-x) = klogx, y(z+x-y) = klogy, z(x+y-z)=klogz
k(ylogx+xlogy)=xy(y+z-x)+xy(z+x-y)= 2xyz
類似的 k(zlogy+ylogz)=k(xlogz+zlog)=2xyz
因此 ylogx+xlogy = zlogy+ylogz = xlogz+zlog
所以 (x^y)(y^x)=(y^z)(z^y)=(z^x)(x^z)
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