Expand the Logarithmic Expression log of (z^3x^2)/( cube root of y^4)

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

Step 1

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

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y^{4}}$ as $y^{\frac{4}{3}}$ .

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

Step 3

Expand $\log (y^{\frac{4}{3}})$ by moving $\frac{4}{3}$ outside the logarithm.

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

Step 4

Combine $\frac{4}{3}$ and $\log (y)$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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