Can anyone help with this?
Thanks in advance!
$\displaystyle \ln\left(\sqrt{x^4\sqrt{y^4\sqrt{z^2}}}\right)$
$\displaystyle \frac{1}{2}\ln\left(x^4\sqrt{y^4\sqrt{z^2}}\right)$
$\displaystyle \frac{1}{2}\ln(x^4) + \ln\left(\sqrt{y^4\sqrt{z^2}}\right)$
$\displaystyle \frac{1}{2}\ln(x^4) + \frac{1}{2}\ln\left(y^4\sqrt{z^2}\right)$
keep going ...
Remember, your properties of log says that $\displaystyle \log_{a}(MN)=\log_{a} M + \log_a{N}$.
To start, I would make your roots exponents. It would be $\displaystyle ln (x^\frac{4}{2} y^\frac{4}{4}z^\frac{2}{8})$.
So using that first property, we would have $\displaystyle ln \ x^\frac{4}{2} + ln\ y^\frac {4}{4} + ln\ z^\frac{2}{8}$.
Another property of logs says that $\displaystyle \log_{a}M^r = r\ log_{a}M$
So we could bring your exponents in front and it would look like
$\displaystyle \frac {4}{2} ln \ x + \frac {4}{4} ln \ y + \frac{2}{8} ln \ z $
You could essentially reduce your fractions, but it is easier to put back together into your original problem with the fractions not reduced.