SOLUTION: {{{x^(log x) = 16(6(x^(log(sqrt(x))))+25)}}} , solve for x log are base 10

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Question 1162787: x%5E%28log+x%29+=+16%286%28x%5E%28log%28sqrt%28x%29%29%29%29%2B25%29 , solve for x log are base 10
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!




Let's simplify the exponent:
                     

So we have:







        Let y = xlog(x)

matrix%282%2C1%2C%22%22%2C%0D%0A%0D%0Ay+=+96y%5E%281%2F2%29%2B400%29

y+=+96sqrt%28y%29%2B400

y-400+=+96sqrt%28y%29

Let 

        z=sqrt%28y%29andz%5E2=y

z%5E2-400+=+96z

z%5E2-96z-400=0

%28z%2B4%29%28z-100%29=0

z+4 = 0;   z-100 = 0
  z = -4;      z = 100

Using

z=sqrt%28y%29

-4=sqrt%28y%29

Impossible with real numbers. 

Using

z=sqrt%28y%29

100=sqrt%28y%29

%28100%29%5E2=%28sqrt%28y%29%29%5E2

10000=y

x%5Elog%28%28x%29%29=+10000

Take logs of both side

log%28%28x%5Elog%28%28x%29%29%29%29=log%28%2810000%29%29

log%28%28x%29%29log%28%28x%29%29=4}

%28log%28%28x%29%29%5E%22%22%29%5E2=4

log%28%28x%29%29=%22%22+%2B-+sqrt%284%29

log%28%28x%29%29+=+%22%22+%2B-+2

x=10%5E%28%22%22+%2B-+2%29

Two solutions:

x=10%5E2=100    x=10%5E%28-2%29=1%5E%22%22%2F10%5E2=1%2F100

Edwin