Trigonometry Examples

$\log_{2} (\sqrt{5}) - \log_{2} (\sqrt{40})$

Step 1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{2} (\frac{\sqrt{5}}{\sqrt{40}})$

Step 2

Combine $\sqrt{5}$ and $\sqrt{40}$ into a single radical.

$\log_{2} (\sqrt{\frac{5}{40}})$

Step 3

Cancel the common factor of $5$ and $40$ .

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Step 3.1

Factor $5$ out of $5$ .

$\log_{2} (\sqrt{\frac{5 (1)}{40}})$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $5$ out of $40$ .

$\log_{2} (\sqrt{\frac{5 \cdot 1}{5 \cdot 8}})$

Step 3.2.2

Cancel the common factor.

$\log_{2} (\sqrt{\frac{5 \cdot 1}{5 \cdot 8}})$

Step 3.2.3

Rewrite the expression.

$\log_{2} (\sqrt{\frac{1}{8}})$

Step 4

Rewrite $\sqrt{\frac{1}{8}}$ as $\frac{\sqrt{1}}{\sqrt{8}}$ .

$\log_{2} (\frac{\sqrt{1}}{\sqrt{8}})$

Step 5

Any root of $1$ is $1$ .

$\log_{2} (\frac{1}{\sqrt{8}})$

Step 6

Simplify the denominator.

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Step 6.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 6.1.1

Factor $4$ out of $8$ .

$\log_{2} (\frac{1}{\sqrt{4 (2)}})$

Step 6.1.2

Rewrite $4$ as $2^{2}$ .

$\log_{2} (\frac{1}{\sqrt{2^{2} \cdot 2}})$

Step 6.2

Pull terms out from under the radical.

$\log_{2} (\frac{1}{2 \sqrt{2}})$

Step 7

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

$\log_{2} (\frac{1}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 8

Combine and simplify the denominator.

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Step 8.1

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

$\log_{2} (\frac{\sqrt{2}}{2 \sqrt{2} \sqrt{2}})$

Step 8.2

Move $\sqrt{2}$ .

$\log_{2} (\frac{\sqrt{2}}{2 (\sqrt{2} \sqrt{2})})$

Step 8.3

Raise $\sqrt{2}$ to the power of $1$ .

$\log_{2} (\frac{\sqrt{2}}{2 ({\sqrt{2}}^{1} \sqrt{2})})$

Step 8.4

Raise $\sqrt{2}$ to the power of $1$ .

$\log_{2} (\frac{\sqrt{2}}{2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})})$

Step 8.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{2} (\frac{\sqrt{2}}{2 {\sqrt{2}}^{1 + 1}})$

Step 8.6

Add $1$ and $1$ .

$\log_{2} (\frac{\sqrt{2}}{2 {\sqrt{2}}^{2}})$

Step 8.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 8.7.1

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

$\log_{2} (\frac{\sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}})$

Step 8.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{2} (\frac{\sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}})$

Step 8.7.3

Combine $\frac{1}{2}$ and $2$ .

$\log_{2} (\frac{\sqrt{2}}{2 \cdot 2^{\frac{2}{2}}})$

Step 8.7.4

Cancel the common factor of $2$ .

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Step 8.7.4.1

Cancel the common factor.

$\log_{2} (\frac{\sqrt{2}}{2 \cdot 2^{\frac{2}{2}}})$

Step 8.7.4.2

Rewrite the expression.

$\log_{2} (\frac{\sqrt{2}}{2 \cdot 2^{1}})$

Step 8.7.5

Evaluate the exponent.

$\log_{2} (\frac{\sqrt{2}}{2 \cdot 2})$

Step 9

Multiply $2$ by $2$ .

$\log_{2} (\frac{\sqrt{2}}{4})$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\log_{2} (\frac{\sqrt{2}}{4})$

Decimal Form:

$- 1.5$