SOLUTION: Find x, x⁵ = 9ˣ

Algebra.Com
Question 1206839: Find x,
x⁵ = 9ˣ

Answer by ikleyn(52778)   (Show Source): You can put this solution on YOUR website!
.
Find x, =
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Your starting equation is 

     = .    (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

     = .     (2)


It is well known fact that the function   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 = 2.71828...


From the other side, the value of   is  2.275598067...


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


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

Solved.

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

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

It allows to plot functions for free automatically.

Print the expression   y = ,   get the plot and compare it with the plot
of horizontal line   y = .


=====================


Post-solution note

    This trick with using monotonicity of the function    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 . . . 



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