Can we use the laws of logs to find $x$ and $y$?

Can we use the laws of logs to find $x$ and $y$?

Question

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.

Suggestion

Which rules do you know to simplify expressions with logarithms? Can you use them to simplify the above equation?

Solution

We will simplify the right hand side of the above equation as far as possible. By using the logarithm quotient rule we get

$$\begin{align*} \log{\frac{4}{9}} - \log{\frac{9}{125}} + 2\log{\frac{9}{5}} & = \log{4} - \log{9} - \log{9} + \log{125} + 2\log{9} - 2\log{5} \\ & = \log{4} + \log{125} - 2\log{5}. \end{align*}$$

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 $y = \frac{1}{5}$.