Solve xy^m=yx^3 for m

  m = 1 + 2log(x)/log(y)

Step-by-step explanation:

Taking logarithms, you have ...

  log(x) +m·log(y) = log(y) +3log(x)

  m·log(y) = log(y) +2·log(x) . . . . subtract log(x)

  m = (log(y) +2·log(x))/log(y) . . . divide by the coefficient of m

  m = 1 +2·log(x)/log(y) . . . . . . . simplify a bit*

* The "simplified" form will depend on your preference. Here, I like the integer 1 brought out because most logs are irrational. The result may be very slightly more accurate if we add 1, rather than log(y)/log(y)--depending on your calculator.