Trigonometry Examples

$\log_{2} (\sqrt[5]{\frac{x y^{4}}{16}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{\frac{x y^{4}}{16}}$ as ${(\frac{x y^{4}}{16})}^{\frac{1}{5}}$ .

$\log_{2} ({(\frac{x y^{4}}{16})}^{\frac{1}{5}})$

Step 2

Expand $\log_{2} ({(\frac{x y^{4}}{16})}^{\frac{1}{5}})$ by moving $\frac{1}{5}$ outside the logarithm.

$\frac{1}{5} \log_{2} (\frac{x y^{4}}{16})$

Step 3

Rewrite $\log_{2} (\frac{x y^{4}}{16})$ as $\log_{2} (x y^{4}) - \log_{2} (16)$ .

$\frac{1}{5} (\log_{2} (x y^{4}) - \log_{2} (16))$

Step 4

Logarithm base $2$ of $16$ is $4$ .

$\frac{1}{5} (\log_{2} (x y^{4}) - 1 \cdot 4)$

Step 5

Multiply $- 1$ by $4$ .

$\frac{1}{5} (\log_{2} (x y^{4}) - 4)$

Step 6

Rewrite $\log_{2} (x y^{4})$ as $\log_{2} (x) + \log_{2} (y^{4})$ .

$\frac{1}{5} (\log_{2} (x) + \log_{2} (y^{4}) - 4)$

Step 7

Expand $\log_{2} (y^{4})$ by moving $4$ outside the logarithm.

$\frac{1}{5} (\log_{2} (x) + 4 \log_{2} (y) - 4)$

Step 8

Apply the distributive property.

$\frac{1}{5} \log_{2} (x) + \frac{1}{5} (4 \log_{2} (y)) + \frac{1}{5} \cdot - 4$

Step 9

Simplify.

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Step 9.1

Combine $\frac{1}{5}$ and $\log_{2} (x)$ .

$\frac{\log_{2} (x)}{5} + \frac{1}{5} (4 \log_{2} (y)) + \frac{1}{5} \cdot - 4$

Step 9.2

Multiply $\frac{1}{5} (4 \log_{2} (y))$ .

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Step 9.2.1

Combine $4$ and $\frac{1}{5}$ .

$\frac{\log_{2} (x)}{5} + \frac{4}{5} \log_{2} (y) + \frac{1}{5} \cdot - 4$

Step 9.2.2

Combine $\frac{4}{5}$ and $\log_{2} (y)$ .

$\frac{\log_{2} (x)}{5} + \frac{4 \log_{2} (y)}{5} + \frac{1}{5} \cdot - 4$

Step 9.3

Combine $\frac{1}{5}$ and $- 4$ .

$\frac{\log_{2} (x)}{5} + \frac{4 \log_{2} (y)}{5} + \frac{- 4}{5}$

Step 10

Move the negative in front of the fraction.

$\frac{\log_{2} (x)}{5} + \frac{4 \log_{2} (y)}{5} - \frac{4}{5}$