Question 1153084
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<pre>

You are given this equation


    {{{x^log(10,(x))}}} = 10.


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


    {{{log(10,(x))*log(10,(x))}}} = 1,

or

    {{{(log(10,(x)))^2}}} = 1.


Take the square root from both sides


    {{{log(10,(x))}}} = +/- 1.


Case 1.   {{{log(10,(x))}}} = 1.


          It implies  x = 10.



Case 2.  {{{log(10,(x))}}} = -1.


         It implies  x = {{{1/10}}}.


<U>ANSWER</U>.  The given equation has 2 (two, TWO) solutions  x = 10  and  x = {{{1/10}}}.

      +------------------------------------------------------------------+
      | The product of the roots (as the question asks) is  10*{{{1/10}}} = 1.   |
      +------------------------------------------------------------------+



To check, I made two plots with different x-scales of the function  y = {{{x^log(10,(x))}}}.  See below.


{{{graph( 330, 330, -2, 20, -10, 20,
          x^log(10,(x)), 10
)}}}


Plot y = {{{x^log(10,(x))}}} (red) and y = 10 (green).



{{{graph( 330, 330, -1, 2, -2, 20,
          x^log(10,(x)), 10
)}}}


Plot y = {{{x^log(10,(x))}}} (red) and y = 10 (green).
</pre>


In the post by &nbsp;@MathLover1, &nbsp;the solution &nbsp;&nbsp;x = {{{1/10}}}  &nbsp;&nbsp;is lost.