Expand the Logarithmic Expression log of (x cube root of y^5)/(z^4)

$\log (\frac{x \sqrt[3]{y^{5}}}{z^{4}})$

Step 1

Rewrite $\log (\frac{x \sqrt[3]{y^{5}}}{z^{4}})$ as $\log (x \sqrt[3]{y^{5}}) - \log (z^{4})$ .

$\log (x \sqrt[3]{y^{5}}) - \log (z^{4})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y^{5}}$ as $y^{\frac{5}{3}}$ .

$\log (x y^{\frac{5}{3}}) - \log (z^{4})$

Step 3

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

$\log (x y^{\frac{5}{3}}) - (4 \log (z))$

Step 4

Multiply $4$ by $- 1$ .

$\log (x y^{\frac{5}{3}}) - 4 \log (z)$

Step 5

Rewrite $\log (x y^{\frac{5}{3}})$ as $\log (x) + \log (y^{\frac{5}{3}})$ .

$\log (x) + \log (y^{\frac{5}{3}}) - 4 \log (z)$

Step 6

Expand $\log (y^{\frac{5}{3}})$ by moving $\frac{5}{3}$ outside the logarithm.

$\log (x) + \frac{5}{3} \log (y) - 4 \log (z)$

Step 7

Combine $\frac{5}{3}$ and $\log (y)$ .

$\log (x) + \frac{5 \log (y)}{3} - 4 \log (z)$