# How do you expand  ln(x/y) - 2ln(x^3) -4lny ?

May 19, 2016

after expanding $- 5 \ln x - 5 \ln y$
after simplication $- \ln {\left(x y\right)}^{5}$

#### Explanation:

$\ln \left(\frac{A}{B}\right) = \ln A - \ln B$
$\ln \left(A B\right) = \ln A + \ln B$
$\ln \left({A}^{B}\right) = B \cdot \ln A$

Using the above two rules we can expand the given expression into:

$\ln x - \ln y - 2 \cdot 3 \cdot \ln x - 4 \ln y$
$\Rightarrow \ln x - \ln y - 6 \ln x - 4 \ln y$
$\mathmr{and} , - 5 \ln x - 5 \ln y$

On further simplification we get

$- 5 \left(\ln x + \ln y\right)$
$\mathmr{and} - 5 \cdot \ln x y$
$\mathmr{and} - \ln {\left(x y\right)}^{5}$