# How do you condense ln 3 − 2 ln 9 + ln 18 ?

Aug 29, 2016

$\ln \left(\frac{2}{3}\right)$

#### Explanation:

Using the $\textcolor{b l u e}{\text{laws of logarithms}} .$These laws apply to logs to any base.

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder}}$

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\log {x}^{n} \Leftrightarrow n \log x} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

$\Rightarrow 2 \ln 9 = \ln {9}^{2} = \ln 81$

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\log x + \log y = \log \left(x y\right)} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

$\Rightarrow \ln 3 + \ln 18 = \ln \left(3 \times 18\right) = \ln 54$

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\log x - \log y = \log \left(\frac{x}{y}\right)} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

$\Rightarrow \ln 54 - \ln 81 = \ln \left(\frac{54}{81}\right) = \ln \left(\frac{2}{3}\right)$
$\textcolor{b l u e}{\text{------------------------------------------------------}}$

$\Rightarrow \ln 3 - 2 \ln 9 + \ln 18 = \ln \left(\frac{2}{3}\right)$