Precalculus Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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Step 1.1.1

Factor $4$ out of $8$ .

$\log_{8} (\frac{1}{\sqrt{4 (2)}})$

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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Step 3.1

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

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

Step 3.2

Move $\sqrt{2}$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Add $1$ and $1$ .

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

Step 3.7

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

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Step 3.7.1

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

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

Step 3.7.2

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

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

Step 3.7.3

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

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

Step 3.7.4

Cancel the common factor of $2$ .

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Step 3.7.4.1

Cancel the common factor.

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

Step 3.7.4.2

Rewrite the expression.

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

Step 3.7.5

Evaluate the exponent.

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

Step 4

Multiply $2$ by $2$ .

$\log_{8} (\frac{\sqrt{2}}{4})$

Step 5

Rewrite $\log_{8} (\frac{\sqrt{2}}{4})$ using the change of base formula.

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Step 5.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 5.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (\frac{\sqrt{2}}{4})}{\log (8)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\frac{\sqrt{2}}{4})}{\log (8)}$

Decimal Form:

$- 0.5$