Expand the Logarithmic Expression log of x/( cube root of z^2y^3)

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify each term.

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

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

Apply the distributive property.

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

Multiply $3$ by $- 1$ .

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