Expand the Logarithmic Expression log of (y^3)/( square root of z^3x^2)

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

Rewrite $\log (\frac{y^{3}}{\sqrt{z^{3}} x^{2}})$ as $\log (y^{3}) - \log (\sqrt{z^{3}} x^{2})$ .

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

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

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

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

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

Rewrite $\log (z^{\frac{3}{2}} x^{2})$ as $\log (z^{\frac{3}{2}}) + \log (x^{2})$ .

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

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

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

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

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

Simplify each term.

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

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

Apply the distributive property.

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

Multiply $2$ by $- 1$ .

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