# How do you condense 4 ln (x+4) - 5 ln x?

Dec 9, 2015

$\ln \left({\left(x + 4\right)}^{4} / {x}^{5}\right)$

#### Explanation:

To condense to a single log, we need to use the properties listed below

Sum to product: $\log A + \log B \iff \log \left(A B\right)$
Difference to quotient: $\log A - \log B \iff \log \left(\frac{A}{B}\right) , B \ne 0$
Power rule: $\log {A}^{n} \iff n \log A$

Given $4 \ln \left(x + 4\right) - 5 \ln x$

Step 1: Reverse the power rule to get
$\ln {\left(x + 4\right)}^{4} - \ln {\left(x\right)}^{5}$

Step 2: Reverse the difference to quotient rule
$\ln \left({\left(x + 4\right)}^{4} / {x}^{5}\right)$

Now, you are done.