Precalculus Examples

$\log_{2} (\sqrt{\frac{2}{64}}) = t$

Step 1

Rewrite the equation as $t = \log_{2} (\sqrt{\frac{2}{64}})$ .

$t = \log_{2} (\sqrt{\frac{2}{64}})$

Step 2

Simplify $\log_{2} (\sqrt{\frac{2}{64}})$ .

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Step 2.1

Cancel the common factor of $2$ and $64$ .

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Step 2.1.1

Factor $2$ out of $2$ .

$t = \log_{2} (\sqrt{\frac{2 (1)}{64}})$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $2$ out of $64$ .

$t = \log_{2} (\sqrt{\frac{2 \cdot 1}{2 \cdot 32}})$

Step 2.1.2.2

Cancel the common factor.

$t = \log_{2} (\sqrt{\frac{2 \cdot 1}{2 \cdot 32}})$

Step 2.1.2.3

Rewrite the expression.

$t = \log_{2} (\sqrt{\frac{1}{32}})$

Step 2.2

Rewrite $\sqrt{\frac{1}{32}}$ as $\frac{\sqrt{1}}{\sqrt{32}}$ .

$t = \log_{2} (\frac{\sqrt{1}}{\sqrt{32}})$

Step 2.3

Any root of $1$ is $1$ .

$t = \log_{2} (\frac{1}{\sqrt{32}})$

Step 2.4

Simplify the denominator.

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Step 2.4.1

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 2.4.1.1

Factor $16$ out of $32$ .

$t = \log_{2} (\frac{1}{\sqrt{16 (2)}})$

Step 2.4.1.2

Rewrite $16$ as $4^{2}$ .

$t = \log_{2} (\frac{1}{\sqrt{4^{2} \cdot 2}})$

Step 2.4.2

Pull terms out from under the radical.

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

Step 2.5

Multiply $\frac{1}{4 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 2.6

Combine and simplify the denominator.

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Step 2.6.1

Multiply $\frac{1}{4 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 2.6.2

Move $\sqrt{2}$ .

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

Step 2.6.3

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

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

Step 2.6.4

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

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

Step 2.6.5

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

$t = \log_{2} (\frac{\sqrt{2}}{4 {\sqrt{2}}^{1 + 1}})$

Step 2.6.6

Add $1$ and $1$ .

$t = \log_{2} (\frac{\sqrt{2}}{4 {\sqrt{2}}^{2}})$

Step 2.6.7

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

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Step 2.6.7.1

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

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

Step 2.6.7.2

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

$t = \log_{2} (\frac{\sqrt{2}}{4 \cdot 2^{\frac{1}{2} \cdot 2}})$

Step 2.6.7.3

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

$t = \log_{2} (\frac{\sqrt{2}}{4 \cdot 2^{\frac{2}{2}}})$

Step 2.6.7.4

Cancel the common factor of $2$ .

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Step 2.6.7.4.1

Cancel the common factor.

$t = \log_{2} (\frac{\sqrt{2}}{4 \cdot 2^{\frac{2}{2}}})$

Step 2.6.7.4.2

Rewrite the expression.

$t = \log_{2} (\frac{\sqrt{2}}{4 \cdot 2^{1}})$

Step 2.6.7.5

Evaluate the exponent.

$t = \log_{2} (\frac{\sqrt{2}}{4 \cdot 2})$

Step 2.7

Multiply $4$ by $2$ .

$t = \log_{2} (\frac{\sqrt{2}}{8})$

Step 3

The result can be shown in multiple forms.

Exact Form:

$t = \log_{2} (\frac{\sqrt{2}}{8})$

Decimal Form:

$t = - 2.5$