# How do you use the properties of logarithms to expand lnsqrt(x^2(x+2))?

Mar 9, 2017

see below

#### Explanation:

Use the following Properties of Logarithm

${\log}_{b} {x}^{n} = n {\log}_{b} x$ and ${\log}_{b} \left(x y\right) = {\log}_{b} x + {\log}_{b} y$

Hence,

ln sqrt(x^2(x+2))=ln(x^2(x+2))^(1/2

$= \frac{1}{2} \cdot \ln \left({x}^{2} \left(x + 2\right)\right)$

$= \frac{1}{2} \left(\ln {x}^{2} + \ln \left(x + 2\right)\right)$

$= \frac{1}{2} \left(2 \ln x + \ln \left(x + 2\right)\right)$

$\therefore = \ln x + \frac{1}{2} \ln \left(x + 2\right)$