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

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

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

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

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

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

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

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

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

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

Combine $\frac{1}{2}$ and $\log (x^{5} y)$ .

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

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

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

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

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

To write $3 \log (z)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $3$ .

$\frac{6 \log (z) - (5 \log (x) + \log (y))}{2}$

Apply the distributive property.

$\frac{6 \log (z) - (5 \log (x)) - \log (y)}{2}$

Multiply $5$ by $- 1$ .

$\frac{6 \log (z) - 5 \log (x) - \log (y)}{2}$