Question 1206839
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Find x,  {{{x^5}}} = {{{9^x}}}
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<pre>
Your starting equation is 

    {{{x^5}}} = {{{9^x}}}.    (1)


Take natural logarithm of both sides

    5*ln(x) = x*ln(9).


Divide both sides by  ln(x)*ln(9).  You will get an EQUIVALENT equation

    {{{x/ln(x)}}} = {{{5/ln(9)}}}.     (2)


It is well known fact that the function {{{x/ln(x)}}}  has the minimum at x = e,
where e is the base of natural logarithms, e = 2.71828...

The minimum of this function at x= e is {{{e/ln(e)}}} = {{{e/1}}} = e = 2.71828...


From the other side, the value of  {{{5/ln(9)}}} is  2.275598067...


THEREFORE, equation (2) has NO solution at x > 0.


<U>ANSWER</U>.  In domain x > 0, the given equation (1) has no solution.
</pre>

Solved.


--------------------


In order to convince yourself visually with the fact that equation &nbsp;(2) 
has no solutions, &nbsp;go to website www.desmos.com/calculator.


It allows to plot functions for free automatically.


Print the expression &nbsp;&nbsp;y = {{{x/ln(x)}}}, &nbsp;&nbsp;get the plot and compare it with the plot
of horizontal line &nbsp;&nbsp;y = {{{5/ln(9)}}}.



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Post-solution note


<pre>
    This trick with using monotonicity of the function  {{{x/ln(x)}}}  is a powerful tool for solving
    many/some exponential-polynomial equations, similar to the given in this post
    or for proving that such equations have no solutions.


    When nothing else does not work, try it . . . 
</pre>