SOLUTION: The product of all the real roots of {{{ x^(logx)=10 }}} is

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Question 1111154: The product of all the real roots of +x%5E%28logx%29=10+ is
Found 2 solutions by ikleyn, MathTherapy:
Answer by ikleyn(54060) About Me  (Show Source):
You can put this solution on YOUR website!
.
+x%5E%28logx%29 = x   for any positive "x",  by the definition of the logarithm.


Therefore, the equation


+x%5E%28logx%29 = 10+ 


is equivalent to equation   x = 10  and has only one solution  x = 10.


So, the answer to your question is 10.

--------------
On logarithms and their properties,  see the lessons
    - WHAT IS the logarithm
    - Properties of the logarithm
    - Change of Base Formula for logarithms
    - Solving logarithmic equations
    - Using logarithms to solve real world problems
in this site.

Also,  you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Logarithms".


Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.



Answer by MathTherapy(10895) About Me  (Show Source):
You can put this solution on YOUR website!
The product of all the real roots of x%5E%28logx%29=10 is
**********************************************
             x%5E%28log+%28x%29%29+=+10, with x > 0
log+%28%28x%5E%28log+%28%28x%29%29%29%29%29+=+log+%28%2810%29%29 ---- Taking log of both sides
   log (x) * log (x) = 1 --- Applying log+%28b%2C+%28a%5Ee%29%29+=+e+%2A+log+%28b%2C+%28a%29%29+, and replacing log (10) with 1
           %28log+%28%28x%29%29%29%5E2+=+1
          sqrt%28log+%28%28x%29%29%5E2%29+=+sqrt%281%29 ---- Taking sqrt of both sides
                 log+%28%28x%29%29+=+0+%2B-+1
               highlight%28x%29+=+10%5E1+=+highlight%2810%29    or     highlight%28x%29+=+10%5E%28-+1%29+=+highlight%281%2F10%29

Product of all the real roots of x%5E%28log+%28%28x%29%29%29+=+10:10%281%2F10%29+=+highlight%281%29