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

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

Step 1

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

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

Step 2

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

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

Step 3

Rewrite $\log (y^{4} z^{\frac{2}{3}})$ as $\log (y^{4}) + \log (z^{\frac{2}{3}})$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify each term.

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

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

Apply the distributive property.

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

Multiply $4$ by $- 1$ .

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