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

Sep 1, 2015

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

#### Explanation:

Use:$\textcolor{w h i t e}{X X} \ln a + \ln b = \ln \left(a b\right)$

and:$\textcolor{w h i t e}{X X} \ln a - \ln b = \ln \left(\frac{a}{b}\right)$

to simplify:

$\ln 3 - 2 \ln 9 + \ln 18$

$= \ln 3 + \ln 18 - \ln {9}^{2}$

$= \ln \left(3 \cdot 18\right) - \ln 81$

$= \ln 54 - \ln 81$

$= \ln \left(\frac{54}{81}\right)$

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

$= \ln 2 - \ln 3$