Expand the Logarithmic Expression log of (z^2)/( cube root of yx^2)

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Rewrite $\log (y^{\frac{1}{3}} x^{2})$ as $\log (y^{\frac{1}{3}}) + \log (x^{2})$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify each term.

Combine $\frac{1}{3}$ and $\log (y)$ .

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

Apply the distributive property.

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

Multiply $2$ by $- 1$ .

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