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

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Multiply $2$ by $- 1$ .

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

Step 5

Rewrite $\log (z^{\frac{4}{3}} y^{4})$ as $\log (z^{\frac{4}{3}}) + \log (y^{4})$ .

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

Step 6

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

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

Step 7

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

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

Step 8

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

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