Laws of logarithms II

There are two more laws of logarithms which we need to be able to use.

There is the addition law:

\log_{a} (x) + \log_{a} (y) = \log_{a} (x y)

and the subtraction law

\log_{a} (x) - \log_{a} (y) = \log_{a} (\frac{x}{y})

It is important to note that we can't directly use these laws if the functions have a coefficient in front. For example,

2 \log (5) - 3 \log (3)

needs to first be changed, using the power law, into

\log (5^{2}) - \log (3^{3})

Now we can use the subtraction law:

\log (\frac{25}{27})

Let's see some more examples!

Assuming $x > 0$ , simplify
$2 \log_{4} (\sqrt{x}) - 3 \log_{4} (\sqrt{x})$
Express
$\log_{u} (\frac{u}{\sqrt{v} w^{3}})$
in terms of $\log_{v} (u)$ and $\log_{v} (w)$ .