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

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify each term.

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

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

Apply the distributive property.

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

Multiply $2$ by $- 1$ .

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