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.

Now as \(125 = 5^3\) we can use the logarithm power rule to get \[\log{4} + \log{5^3} - 2\log{5} = \log{4} + 3\log{5} - 2\log{5} = \log{4}+\log{5}=\log{20}.\] Therefore, we get \(\log{x} = \log{20}\) and thus \(x = 20\).

Now for \(y\) we get the equation \(2 + \log{y} = \log{20}\). As all logarithms are to base \(10\) we can write \(2 = \log{100}\) and therefore \[\begin{align*} \log{y} & = \log{20} - \log{100} \\ & = \log{\frac{20}{100}} \\ & = \log{\frac{1}{5}}. \end{align*}\]Hence \(y = \frac{1}{5}\).