# How do you express in terms of ln2 and ln3 given ln 3 sqrt 2?

May 4, 2016

$\ln \left(3 \sqrt{2}\right) = \ln \left(3\right) + \frac{1}{2} \ln \left(2\right)$

#### Explanation:

Using the properties that $\ln \left(a b\right) = \ln \left(a\right) + \ln \left(b\right)$ and $\ln \left({a}^{x}\right) = x \ln \left(a\right)$, we have:

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

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

$= \ln \left(3\right) + \frac{1}{2} \ln \left(2\right)$