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

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

Step 1

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

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y^{4} x^{3}}$ as ${(y^{4} x^{3})}^{\frac{1}{3}}$ .

$\log (z^{4}) - \log ({(y^{4} x^{3})}^{\frac{1}{3}})$

Step 3

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

$4 \log (z) - \log ({(y^{4} x^{3})}^{\frac{1}{3}})$

Step 4

Expand $\log ({(y^{4} x^{3})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$4 \log (z) - (\frac{1}{3} \log (y^{4} x^{3}))$

Step 5

Combine $\frac{1}{3}$ and $\log (y^{4} x^{3})$ .

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

Step 6

Rewrite $\log (y^{4} x^{3})$ as $\log (y^{4}) + \log (x^{3})$ .

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

Step 7

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

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

Step 8

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

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

Step 9

To write $4 \log (z)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 10

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

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

Step 11

Combine the numerators over the common denominator.

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

Step 12

Simplify the numerator.

Multiply $3$ by $4$ .

$\frac{12 \log (z) - (4 \log (y) + 3 \log (x))}{3}$

Apply the distributive property.

$\frac{12 \log (z) - (4 \log (y)) - (3 \log (x))}{3}$

Multiply $4$ by $- 1$ .

$\frac{12 \log (z) - 4 \log (y) - (3 \log (x))}{3}$

Multiply $3$ by $- 1$ .

$\frac{12 \log (z) - 4 \log (y) - 3 \log (x)}{3}$