Trigonometry Examples

$\log_{2} (\sqrt{10 x})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 x}$ as ${(10 x)}^{\frac{1}{2}}$ .

$\log_{2} ({(10 x)}^{\frac{1}{2}})$

Step 2

Expand $\log_{2} ({(10 x)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{2} (10 x)$

Step 3

Rewrite $\log_{2} (10 x)$ as $\log_{2} (10) + \log_{2} (x)$ .

$\frac{1}{2} (\log_{2} (10) + \log_{2} (x))$

Step 4

Simplify each term.

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Rewrite $\log_{2} (10)$ as $\log_{2} (2 (5))$ .

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

Rewrite $\log_{2} (2 (5))$ as $\log_{2} (2) + \log_{2} (5)$ .

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

Logarithm base $2$ of $2$ is $1$ .

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

Step 5

Apply the distributive property.

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

Step 6

Simplify.

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Multiply $\frac{1}{2}$ by $1$ .

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

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

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

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

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