SOLUTION: Find x, 3ˣ = x⁹

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Question 1206837: Find x,
3ˣ = x⁹
Found 3 solutions by Edwin McCravy, MathLover1, ikleyn:
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!

3%5Ex=x%5E9

If in an equation to solve, the variable to solve for appears both as an
exponent and a non-exponent, as it does here, then there is no way to solve for
the variable other than by iterative methods.  

Iterative methods are trial and error processes.  
Try a number. If it's too big, try a smaller number.
If it's too small try a bigger number.
Always try a number between the last number you tried that was too big, and
the last number you tried that was too small. 
Eventually you will have the solution trapped between two numbers, that get
closer and closer together.

That's a very time-consuming process, but technology can do that trial and 
error process instantaneously.

On a TI-84 Plus CE calculator you can get x = 1.1508248, an approximate solution
trapped between 1.15082475 and 1.15082485.

Edwin

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

Find x,

3%5Ex+=+x%5E9..... raise both sides to the power of 1%2Fx

3%5E%28x%2Fx%29+=+x%5E%289%2Fx%29

3%5E1+=+x%5E%289%2Fx%29...we can write the power of 1 as 3%281%2F3%29

3%5E%283%281%2F3%29%29+=+x%5E%289%2Fx%29

%283%5E3%29%5E%281%2F3%29+=+x%5E%289%2Fx%29

27%5E%281%2F3%29+=+x%5E%289%2Fx%29........ raise both sides to the power of 1%2F9

%2827%5E%281%2F3%29%29%5E%281%2F9%29+=+%28x%5E%289%2Fx%29%29%5E%281%2F9%29

27%5E%281%2F27%29+=+x%5E%281%2Fx%29

for left side to be equal to the right side, x+must be equal to 27

so, x=27

check:
3%5E27+=+27%5E9
7625597484987=7625597484987


Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
.
Find x, 3%5Ex = x%5E9.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

Your starting equation is 

    3%5Ex = x%5E9.    (1)


Take natural logarithm of both sides

    x*ln(3) = 9*ln(x).


Divide both sides by ln(x)*ln(3).  You will get

    x%2Fln%28x%29 = 9%2Fln%283%29.


Transform right side equivalently multiplying by 1 = 3%2F3

    x%2Fln%28x%29 = %283%2F3%29%2A%289%2Fln%283%29%29 = %283%2A9%29%2F%283%2Aln%283%29%29 = 27%2Fln%283%5E3%29 = 27%2Fln%2827%29.


Thus we have this equation 

    x%2Fln%28x%29 = 27%2Fln%2827%29.    (2)


It is well known fact that if x > e, then x%2Fln%28x%29 is monotonically increasing function.
Here e is the base of natural logarithms, e = 2.71818...
THEREFORE, from equation (2), we conclude that x = 27 is the unique solution in domain x > e.


ANSWER.  In domain x > e, the unique solution to equation (1) is  x= 27.


CHECK.  At x= 27, left side of equation (1) is 3%5E27;

                  right side of equation (1) is 27%5E9 = %283%5E3%29%5E9 = 3%5E27.

        Thus, at x= 27, both sides of equation (1) are equal.

Solved.

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

Post-solution notes 

    1. The original equation has the second solution in the domain  1 < x < e, where the function x%2Fln%28x%29 
       is also monotonic; but this solution is not an integer number.
       This second solution can be found numerically as an approximate value.


    2. This trick with using monotonicity of the function x%2Fln%28x%29 is a powerful tool for solving
       many/some exponential-polynomial equations, similar to the given in this post.