Introduce new variable  x = .


Then the equation takes the form


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



The roots of the polynomial  (approximate values)  are:


     =  3.72008   

     = −4.22409

     = −0.248 + 3.9246i

     = −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 = ,  for us only the positive real value 

     =  3.72008

make sense.


Then from   = 3.72008   you have   =  = 0.26881.


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


    m =  = -1.19583   (approximate).    ANSWER