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

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify each term.

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

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

Apply the distributive property.

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

Multiply $2$ by $- 1$ .

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