This doesn't prove anything, unfortunately. It doesn't even make use of the fact that m,n are integers. If m,n are not restricted to integers, there exists an infinite number of solutions (e.g. m=8.043, n=1.46).

The mistake in your proof is that you considered the shape of the log function while implicitly holding m on the left-hand-side constant (i.e. you should have concluded "for a constant m", there is only 1 n satisfying this equality"). However, since m is variable, you have to consider an infinite number of log functions. If I missed something, let me know.