Question 1176011
.



            The  "solution"  in the post by @CubeyThePenguin is  TOTALLY  WRONG.


            It has nothing in common with the correct solution.



<pre>
Introduce new variable  x = {{{(1/3)^m}}}.


Then the equation takes the form


    x^4 + x^3 - 243 = 0.



The roots of the polynomial  (approximate values)  are:


    {{{x[1]}}} =  3.72008   

    {{{x[2]}}} = −4.22409

    {{{x[3]}}} = −0.248 + 3.9246i

    {{{x[4]}}} = −0.248 − 3.9246i



This polynomial has no rational roots that can be found using Rational Root Test.

Roots were found using quartic formulas.



Of these roots for x = {{{(1/3)^m}}},  for us only the positive real value 

    {{{x[1]}}} =  3.72008

make sense.


Then from  {{{(1/3)^m}}} = 3.72008   you have  {{{3^m}}} = {{{1/3.72008}}} = 0.26881.


And THEREFORE,  m*log(3) = log(0.26881), which implies


    m = {{{log((0.26881))/log((3))}}} = -1.19583   (approximate).    <U>ANSWER</U>
</pre>

Solved.