# How do you use the properties of logarithms to expand lnroot4(x^3(x^2+3))?

Aug 28, 2017

Given: $\ln \left(\sqrt[4]{{x}^{3} \left({x}^{2} + 3\right)}\right)$

The root 4 can be written as the $\frac{1}{4}$ power:
$\ln \left({\left({x}^{3} \left({x}^{2} + 3\right)\right)}^{\frac{1}{4}}\right)$

The property $\ln \left({a}^{c}\right) = c \ln \left(a\right)$ tells us that the $\frac{1}{4}$ comes outside as multiplication:

$\frac{1}{4} \ln \left({x}^{3} \left({x}^{2} + 3\right)\right)$

Use the property $\ln \left(u v\right) = \ln \left(u\right) + \ln \left(v\right)$ to make the two factors within the argument become the sum of two logarithms but both are still multiplied by $\frac{1}{4}$:

$\frac{1}{4} \ln \left({x}^{3}\right) + \frac{1}{4} \ln \left({x}^{2} + 3\right)$

Use the property $\ln \left({a}^{c}\right) = c \ln \left(a\right)$ to bring the cube outside as multiplication by 3:

$\frac{3}{4} \ln \left(x\right) + \frac{1}{4} \ln \left({x}^{2} + 3\right)$