# How do you expand  ln sqrt (( x^2( x+2))?

May 30, 2016

For this problem, you must remember that $\sqrt{x} = {x}^{\frac{1}{2}}$

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

Using the product rule (${\log}_{a} n + {\log}_{a} m = {\log}_{a} \left(n \times m\right)$) and distributing.

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

Using exponent rules to simplify ((x^2)^(1/2) =x^ (2 xx 1/2) = x^1)
$= \ln x + \ln {\left(x + 2\right)}^{\frac{1}{2}}$

Now use the log rule $\log {n}^{a} = a \log n$

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

Hopefully this helps!