# How do you rewrite this Logarithmic problem in expanded form?

May 11, 2018

$= \frac{7}{2} \log x - \frac{5}{2} \log y - \frac{7}{2} \log z$

#### Explanation:

$\log \sqrt{{x}^{7} / \left({y}^{5} {z}^{7}\right)} = \log {\left({x}^{7} / \left({y}^{5} {z}^{7}\right)\right)}^{\frac{1}{2}}$

Recall that $\log \left({x}^{a}\right) = a \log x$, so

$\log {\left({x}^{7} / \left({y}^{5} {z}^{7}\right)\right)}^{\frac{1}{2}} = \frac{1}{2} \log \left({x}^{7} / \left({y}^{5} {z}^{7}\right)\right)$

Furthermore, recalling that $\log \left(\frac{a}{b}\right) = \log a - \log b ,$

$\frac{1}{2} \log \left({x}^{7} / \left({y}^{5} {z}^{7}\right)\right) = \frac{1}{2} \left[\log \left({x}^{7}\right) - \log \left({y}^{5} {z}^{7}\right)\right]$

Recalling that $\log \left(a b\right) = \log a + \log b ,$

$\frac{1}{2} \left[\log \left({x}^{7}\right) - \log \left({y}^{5} {z}^{7}\right)\right] = \frac{1}{2} \left[\log \left({x}^{7}\right) - \left(\log \left({y}^{5}\right) + \log \left({z}^{7}\right)\right)\right]$

Apply the exponent property to all remaining logarithms and distribute the negative through:

$= \frac{1}{2} \left[7 \log x - 5 \log \left(y\right) - 7 \log \left(z\right)\right]$

Distribute the $\frac{1}{2} :$

$= \frac{7}{2} \log x - \frac{5}{2} \log y - \frac{7}{2} \log z$