Trigonometry Examples

$\log_{8} (\sqrt[3]{x y^{4}})$

Step 1

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

$\log_{8} ({(x y^{4})}^{\frac{1}{3}})$

Step 2

Expand $\log_{8} ({(x y^{4})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log_{8} (x y^{4})$

Step 3

Rewrite $\log_{8} (x y^{4})$ as $\log_{8} (x) + \log_{8} (y^{4})$ .

$\frac{1}{3} (\log_{8} (x) + \log_{8} (y^{4}))$

Step 4

Expand $\log_{8} (y^{4})$ by moving $4$ outside the logarithm.

$\frac{1}{3} (\log_{8} (x) + 4 \log_{8} (y))$

Step 5

Apply the distributive property.

$\frac{1}{3} \log_{8} (x) + \frac{1}{3} (4 \log_{8} (y))$

Step 6

Combine $\frac{1}{3}$ and $\log_{8} (x)$ .

$\frac{\log_{8} (x)}{3} + \frac{1}{3} (4 \log_{8} (y))$

Step 7

Multiply $\frac{1}{3} (4 \log_{8} (y))$ .

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Step 7.1

Combine $4$ and $\frac{1}{3}$ .

$\frac{\log_{8} (x)}{3} + \frac{4}{3} \log_{8} (y)$

Step 7.2

Combine $\frac{4}{3}$ and $\log_{8} (y)$ .

$\frac{\log_{8} (x)}{3} + \frac{4 \log_{8} (y)}{3}$