SOLUTION: The product of all the real roots of x^log(base 10)x = 10 is

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Question 1153084: The product of all the real roots of x^log(base 10)x = 10 is
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20855) About Me  (Show Source):
You can put this solution on YOUR website!

x%5E%28log%2810%2Cx%29+%29=+10......take log of both sides
log%28x%5E%28log%28x%29+%29%29=+log%2810%29
log%28x%29%2Alog%28x%29=+1
log%5E2%28x%29=+1
log%28x%29=+sqrt%281%29
solutions for this log:
log%28x%29=+1 or log%28x%29=+-1 -> since log, disregard negative solution
log%28x%29+=+1........since log%2810%29=1
log%28x%29+=+log%2810%29
x=10
since you have only one+solution, product is 1%2A10=10

Answer by ikleyn(53763) About Me  (Show Source):
You can put this solution on YOUR website!
.

You are given this equation


    x%5Elog%2810%2C%28x%29%29 = 10.


Take log(base 10) from both sides.  You will get


    log%2810%2C%28x%29%29%2Alog%2810%2C%28x%29%29 = 1,

or

    %28log%2810%2C%28x%29%29%29%5E2 = 1.


Take the square root from both sides


    log%2810%2C%28x%29%29 = +/- 1.


Case 1.   log%2810%2C%28x%29%29 = 1.


          It implies  x = 10.



Case 2.  log%2810%2C%28x%29%29 = -1.


         It implies  x = 1%2F10.


ANSWER.  The given equation has 2 (two, TWO) solutions  x = 10  and  x = 1%2F10.

      +------------------------------------------------------------------+
      | The product of the roots (as the question asks) is  10*1%2F10 = 1.   |
      +------------------------------------------------------------------+



To check, I made two plots with different x-scales of the function  y = x%5Elog%2810%2C%28x%29%29.  See below.





Plot y = x%5Elog%2810%2C%28x%29%29 (red) and y = 10 (green).






Plot y = x%5Elog%2810%2C%28x%29%29 (red) and y = 10 (green).


In the post by  @MathLover1,  the solution   x = 1%2F10   is lost.