How do you expand ln(x(x+11))^3?

Feb 9, 2016

$\ln {\left(x \left(x + 11\right)\right)}^{3} = 3 \ln \left(x\right) + 3 \ln \left(x + 1\right)$

Explanation:

You can use the following two logarithmic laws:

[1] $\text{ } {\log}_{a} \left(m \cdot n\right) = {\log}_{a} \left(m\right) + {\log}_{a} \left(n\right)$

[2] $\text{ } {\log}_{a} \left({m}^{r}\right) = r \cdot {\log}_{a} \left(m\right)$

$\ln {\left(x \left(x + 11\right)\right)}^{3} \stackrel{\text{[2] }}{=} 3 \cdot \ln \left(x \left(x + 1\right)\right)$
$\stackrel{\text{[1] }}{=} 3 \left(\ln \left(x\right) + \ln \left(x + 1\right)\right)$
$= 3 \ln \left(x\right) + 3 \ln \left(x + 1\right)$