Expand the Logarithmic Expression log base w of ((x^2-6)^4)/( cube root of x^2+8)

$\log_{w} (\frac{{(x^{2} - 6)}^{4}}{\sqrt[3]{x^{2} + 8}})$

Step 1

Rewrite $\log_{w} (\frac{{(x^{2} - 6)}^{4}}{\sqrt[3]{x^{2} + 8}})$ as $\log_{w} ({(x^{2} - 6)}^{4}) - \log_{w} (\sqrt[3]{x^{2} + 8})$ .

$\log_{w} ({(x^{2} - 6)}^{4}) - \log_{w} (\sqrt[3]{x^{2} + 8})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2} + 8}$ as ${(x^{2} + 8)}^{\frac{1}{3}}$ .

$\log_{w} ({(x^{2} - 6)}^{4}) - \log_{w} ({(x^{2} + 8)}^{\frac{1}{3}})$

Step 3

Expand $\log_{w} ({(x^{2} - 6)}^{4})$ by moving $4$ outside the logarithm.

$4 \log_{w} (x^{2} - 6) - \log_{w} ({(x^{2} + 8)}^{\frac{1}{3}})$

Step 4

Expand $\log_{w} ({(x^{2} + 8)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$4 \log_{w} (x^{2} - 6) - (\frac{1}{3} \log_{w} (x^{2} + 8))$

Step 5

Combine $\frac{1}{3}$ and $\log_{w} (x^{2} + 8)$ .

$4 \log_{w} (x^{2} - 6) - \frac{\log_{w} (x^{2} + 8)}{3}$

Step 6

To write $4 \log_{w} (x^{2} - 6)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$4 \log_{w} (x^{2} - 6) \cdot \frac{3}{3} - \frac{\log_{w} (x^{2} + 8)}{3}$

Step 7

Combine $4 \log_{w} (x^{2} - 6)$ and $\frac{3}{3}$ .

$\frac{4 \log_{w} (x^{2} - 6) \cdot 3}{3} - \frac{\log_{w} (x^{2} + 8)}{3}$

Step 8

Combine the numerators over the common denominator.

$\frac{4 \log_{w} (x^{2} - 6) \cdot 3 - \log_{w} (x^{2} + 8)}{3}$

Step 9

Multiply $3$ by $4$ .

$\frac{12 \log_{w} (x^{2} - 6) - \log_{w} (x^{2} + 8)}{3}$