# How do you condense ln x-4 ln(x + 2) + 3 ln x?

Oct 14, 2016

$\ln x - 4 \ln \left(x + 2\right) + 3 \ln x = \ln {\left(\frac{x}{x + 2}\right)}^{4}$

#### Explanation:

Some of the basic operations in logarithm are ${\log}_{m} a + {\log}_{m} b = {\log}_{m} \left(a b\right)$, ${\log}_{m} a - {\log}_{m} b = {\log}_{m} \left(\frac{a}{b}\right)$ and $n {\log}_{m} a = {\log}_{m} \left({a}^{n}\right)$. Further $\ln$ is special form where $m = e$ i.e. base is $e$. Using them

$\ln x - 4 \ln \left(x + 2\right) + 3 \ln x$

= $\ln x - \ln {\left(x + 2\right)}^{4} + \ln {x}^{3}$

= ln((x×x^3)/(x+2)^4)

= $\ln \left({x}^{4} / {\left(x + 2\right)}^{4}\right)$

= $\ln {\left(\frac{x}{x + 2}\right)}^{4}$