If \[\log{x} = 2 + \log{y} = \log{\frac{4}{9}} - \log{\frac{9}{125}} + 2\log{\frac{9}{5}},\] where all logarithms are to base \(10\), find \(x\) and \(y\) without using tables.

If \[\log{x} = 2 + \log{y} = \log{\frac{4}{9}} - \log{\frac{9}{125}} + 2\log{\frac{9}{5}},\] where all logarithms are to base \(10\), find \(x\) and \(y\) without using tables.