Expand the Logarithmic Expression log of ( square root of yz^5)/(x^3)

$\log (\frac{\sqrt{y z^{5}}}{x^{3}})$

Rewrite $\log (\frac{\sqrt{y z^{5}}}{x^{3}})$ as $\log (\sqrt{y z^{5}}) - \log (x^{3})$ .

$\log (\sqrt{y z^{5}}) - \log (x^{3})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y z^{5}}$ as ${(y z^{5})}^{\frac{1}{2}}$ .

$\log ({(y z^{5})}^{\frac{1}{2}}) - \log (x^{3})$

Expand $\log ({(y z^{5})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (y z^{5}) - \log (x^{3})$

Expand $\log (x^{3})$ by moving $3$ outside the logarithm.

$\frac{1}{2} \log (y z^{5}) - (3 \log (x))$

Multiply $3$ by $- 1$ .

$\frac{1}{2} \log (y z^{5}) - 3 \log (x)$

Rewrite $\log (y z^{5})$ as $\log (y) + \log (z^{5})$ .

$\frac{1}{2} (\log (y) + \log (z^{5})) - 3 \log (x)$

Expand $\log (z^{5})$ by moving $5$ outside the logarithm.

$\frac{1}{2} (\log (y) + 5 \log (z)) - 3 \log (x)$

Simplify each term.

Apply the distributive property.

$\frac{1}{2} \log (y) + \frac{1}{2} (5 \log (z)) - 3 \log (x)$

Combine $\frac{1}{2}$ and $\log (y)$ .

$\frac{\log (y)}{2} + \frac{1}{2} (5 \log (z)) - 3 \log (x)$

Multiply $\frac{1}{2} (5 \log (z))$ .

Combine $5$ and $\frac{1}{2}$ .

$\frac{\log (y)}{2} + \frac{5}{2} \log (z) - 3 \log (x)$

Combine $\frac{5}{2}$ and $\log (z)$ .

$\frac{\log (y)}{2} + \frac{5 \log (z)}{2} - 3 \log (x)$