1. ## Logs

Use Logs to expand the expression

2. Originally Posted by tmac11522
Use Logs to expand the expression

I take it by expand using logs, you mean use log laws to simplify...

Notice

$\sqrt{z} = z^{\frac{1}{2}}$

$\sqrt{y^2\sqrt{z}} = \sqrt{y^2z^{\frac{1}{2}}} = (y^2z^{\frac{1}{2}})^{\frac{1}{2}}$

$\sqrt{x^4\sqrt{y^2\sqrt{z}}} = \sqrt{x^4(y^2z^{\frac{1}{2}})^{\frac{1}{2}}} = (x^4(y^2z^{\frac{1}{2}})^{\frac{1}{2}})^{\frac{1}{ 2}}$.

So $\ln{(\sqrt{x^4\sqrt{y^2\sqrt{z}}})} = \ln{((x^4(y^2z^{\frac{1}{2}})^{\frac{1}{2}})^{\fra c{1}{2}})}$

$= \frac{1}{2}\ln{x^4(y^2z^{\frac{1}{2}})^{\frac{1}{2 }}}$

$= \frac{1}{2}(\ln{x^4} + \ln{(y^2z^{\frac{1}{2}})^{\frac{1}{2}}}$

$= \frac{1}{2}(\ln{x^4} + \frac{1}{2}\ln{y^2z^{\frac{1}{2}}})$

$= \frac{1}{2}[4\ln{x} + \frac{1}{2}(\ln{y^2} + \ln{z^{\frac{1}{2}}})]$

$= \frac{1}{2}[4\ln{x} + \frac{1}{2}(2\ln{y} + \frac{1}{2}\ln{z})]$

$= \frac{1}{2}(4\ln{x} + \ln{y} + \frac{1}{4}\ln{z})$

$= 2\ln{x} + \frac{1}{2}\ln{y} + \frac{1}{8}\ln{z}$.