# How do you expand log (10/y)?

Mar 7, 2016

$\log 10 - \log y$
The second law of logs says tha $\log \textcolor{red}{x} - \log \textcolor{b l u e}{y}$ can be simplified to $\log \left(\frac{\textcolor{red}{x}}{\textcolor{b l u e}{y}}\right)$. The only requirement for this to work is that both $\log$s must have the same bases.
For our problem of $\log \left(\frac{10}{y}\right)$, we have to expand the expression. We need make sure that we end up with two $\log$s with the same bases.
The first step is also the last step: take the numerator, and make that the first part of the expression, and make the denominator the second part. What I mean is this: $\log \left(\frac{\textcolor{red}{10}}{\textcolor{b l u e}{y}}\right)$ becomes $\log \textcolor{red}{10} \textcolor{\mathmr{and} a n \ge}{-} \log \textcolor{b l u e}{y}$. The $\textcolor{\mathmr{and} a n \ge}{-}$ is very important, because that is what differentiates division from multiplication.