Expand the Logarithmic Expression log of (y^2 cube root of z^5)/(x^4)

$\log (\frac{y^{2} \sqrt[3]{z^{5}}}{x^{4}})$

Step 1

Rewrite $\log (\frac{y^{2} \sqrt[3]{z^{5}}}{x^{4}})$ as $\log (y^{2} \sqrt[3]{z^{5}}) - \log (x^{4})$ .

$\log (y^{2} \sqrt[3]{z^{5}}) - \log (x^{4})$

Step 2

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

$\log (y^{2} z^{\frac{5}{3}}) - \log (x^{4})$

Step 3

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

$\log (y^{2} z^{\frac{5}{3}}) - (4 \log (x))$

Step 4

Multiply $4$ by $- 1$ .

$\log (y^{2} z^{\frac{5}{3}}) - 4 \log (x)$

Step 5

Rewrite $\log (y^{2} z^{\frac{5}{3}})$ as $\log (y^{2}) + \log (z^{\frac{5}{3}})$ .

$\log (y^{2}) + \log (z^{\frac{5}{3}}) - 4 \log (x)$

Step 6

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

$2 \log (y) + \log (z^{\frac{5}{3}}) - 4 \log (x)$

Step 7

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

$2 \log (y) + \frac{5}{3} \log (z) - 4 \log (x)$

Step 8

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

$2 \log (y) + \frac{5 \log (z)}{3} - 4 \log (x)$