Expand the Logarithmic Expression log of ( cube root of x^7)/(y^2z)

$\log (\frac{\sqrt[3]{x^{7}}}{y^{2} z})$

Step 1

Rewrite $\log (\frac{\sqrt[3]{x^{7}}}{y^{2} z})$ as $\log (\sqrt[3]{x^{7}}) - \log (y^{2} z)$ .

$\log (\sqrt[3]{x^{7}}) - \log (y^{2} z)$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{7}}$ as $x^{\frac{7}{3}}$ .

$\log (x^{\frac{7}{3}}) - \log (y^{2} z)$

Step 3

Expand $\log (x^{\frac{7}{3}})$ by moving $\frac{7}{3}$ outside the logarithm.

$\frac{7}{3} \log (x) - \log (y^{2} z)$

Step 4

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

$\frac{7}{3} \log (x) - (\log (y^{2}) + \log (z))$

Step 5

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

$\frac{7}{3} \log (x) - (2 \log (y) + \log (z))$

Step 6

Simplify each term.

Combine $\frac{7}{3}$ and $\log (x)$ .

$\frac{7 \log (x)}{3} - (2 \log (y) + \log (z))$

Apply the distributive property.

$\frac{7 \log (x)}{3} - (2 \log (y)) - \log (z)$

Multiply $2$ by $- 1$ .

$\frac{7 \log (x)}{3} - 2 \log (y) - \log (z)$