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

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3} y z^{5}}$ as ${(x^{3} y z^{5})}^{\frac{1}{2}}$ .

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

Expand $\log ({(x^{3} y z^{5})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

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

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

Rewrite $\log (x^{3} y)$ as $\log (x^{3}) + \log (y)$ .

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

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

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

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

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

Apply the distributive property.

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

Simplify.

Multiply $\frac{1}{2} (3 \log (x))$ .

Combine $3$ and $\frac{1}{2}$ .

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

Combine $\frac{3}{2}$ and $\log (x)$ .

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

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

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

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

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

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

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

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