I think this might be a contest math question, so I'm tagging it as such.
I don't know how to do something like this by hand (or if it's even possible, though I would presume it is if it's from a contest exam). I wrote a script in Mathematica to determine the answer. Here's the code:
t = Table[6^x - 5^y, {x, 1, 200}, {y, 1, 200}];
For[m = 1, m <= 200, m++,
For[n = 1, n <= 200, n++,
If[t[[m, n]] == %1, Print[{m, n}]];
];
];
(where $\%1$ denotes the computed product, approximately $6.33\times10^{49}$) which returns $\{64,64\}$, and so $x-y=0$. How would I do this without the aid of software?