Are o,a,b real numbers? Do you know for certain that these kinds of equations can be solved by hand? If not, they most likely aren't solveable except by numerical approximation.

You would use functions such as Solve to do this. When the algebra is untractable, you use NSolve. This is most likely the case here, since this doesn't give us a symbolic answer.

Solve[{1/x + 1/y == o, x Log[a/y] == y Log[b/x], 1/a + 1/b == 1}, {x, y}]

For specific values of your parameters, you may be able to get values for NSolve. You'd want to confirm that there's only one unique solution first or prime NSolve so that it finds the solution you are looking for.