Linear equations

By (1) :
log x = ylogy = log y^y , i.e. x = y^y....(3)
By (2) :
log y = xlogx = log x^x , i.e. y = x^x....(4)
Sub (4) to (3) :
x = (x^x)^(x^x) = x^[x^(x+1)]
1 = x^(x+1)
log 1 = (x+1) log x
0 = (x + 1) (log x)
x = - 1 or x = 1
y = 1^1 = 1 or (-1)^(-1) = - 1


2010-03-19 19:33:19 補充：
1 = x^(x+1),

when x <> 1 :

x ^ 0 = x ^ (x+1)

0 = x + 1

x = - 1

;

2010-03-19 19:34:44 補充：
when x = 1 ,

1 = 1^(1+1)

so x can be 1

2010-03-19 19:35:59 補充：
上面take log 有不嚴謹。

log (- 1) 不妥。

2010-03-19 19:37:36 補充：
x = y = 1

or

x = y = - 1

0 = (x + 1) (log x)
之後係點架??