Expand the Logarithmic Expression log of ( cube root of y^4)/(xz^2)

$\log (\frac{\sqrt[3]{y^{4}}}{x z^{2}})$

Step 1

Rewrite $\log (\frac{\sqrt[3]{y^{4}}}{x z^{2}})$ as $\log (\sqrt[3]{y^{4}}) - \log (x z^{2})$ .

$\log (\sqrt[3]{y^{4}}) - \log (x z^{2})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y^{4}}$ as $y^{\frac{4}{3}}$ .

$\log (y^{\frac{4}{3}}) - \log (x z^{2})$

Step 3

Expand $\log (y^{\frac{4}{3}})$ by moving $\frac{4}{3}$ outside the logarithm.

$\frac{4}{3} \log (y) - \log (x z^{2})$

Step 4

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

$\frac{4}{3} \log (y) - (\log (x) + \log (z^{2}))$

Step 5

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

$\frac{4}{3} \log (y) - (\log (x) + 2 \log (z))$

Step 6

Simplify each term.

Combine $\frac{4}{3}$ and $\log (y)$ .

$\frac{4 \log (y)}{3} - (\log (x) + 2 \log (z))$

Apply the distributive property.

$\frac{4 \log (y)}{3} - \log (x) - (2 \log (z))$

Multiply $2$ by $- 1$ .

$\frac{4 \log (y)}{3} - \log (x) - 2 \log (z)$