Precalculus Examples

$\log_{6} (\sqrt{\frac{1}{6}})$

Step 1

Rewrite $\sqrt{\frac{1}{6}}$ as $\frac{\sqrt{1}}{\sqrt{6}}$ .

$\log_{6} (\frac{\sqrt{1}}{\sqrt{6}})$

Step 2

Any root of $1$ is $1$ .

$\log_{6} (\frac{1}{\sqrt{6}})$

Step 3

Multiply $\frac{1}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\log_{6} (\frac{1}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}})$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{1}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\log_{6} (\frac{\sqrt{6}}{\sqrt{6} \sqrt{6}})$

Step 4.2

Raise $\sqrt{6}$ to the power of $1$ .

$\log_{6} (\frac{\sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}})$

Step 4.3

Raise $\sqrt{6}$ to the power of $1$ .

$\log_{6} (\frac{\sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}})$

Step 4.4

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

$\log_{6} (\frac{\sqrt{6}}{{\sqrt{6}}^{1 + 1}})$

Step 4.5

Add $1$ and $1$ .

$\log_{6} (\frac{\sqrt{6}}{{\sqrt{6}}^{2}})$

Step 4.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\log_{6} (\frac{\sqrt{6}}{{(6^{\frac{1}{2}})}^{2}})$

Step 4.6.2

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

$\log_{6} (\frac{\sqrt{6}}{6^{\frac{1}{2} \cdot 2}})$

Step 4.6.3

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

$\log_{6} (\frac{\sqrt{6}}{6^{\frac{2}{2}}})$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$\log_{6} (\frac{\sqrt{6}}{6^{\frac{2}{2}}})$

Step 4.6.4.2

Rewrite the expression.

$\log_{6} (\frac{\sqrt{6}}{6^{1}})$

Step 4.6.5

Evaluate the exponent.

$\log_{6} (\frac{\sqrt{6}}{6})$

Step 5

Rewrite $\frac{\sqrt{6}}{6}$ as $\sqrt{6} \cdot 6^{- 1}$ .

$\log_{6} (\sqrt{6} \cdot 6^{- 1})$

Step 6

Rewrite $\log_{6} (\sqrt{6} \cdot 6^{- 1})$ as $\log_{6} (\sqrt{6}) + \log_{6} (6^{- 1})$ .

$\log_{6} (\sqrt{6}) + \log_{6} (6^{- 1})$

Step 7

Use logarithm rules to move $- 1$ out of the exponent.

$\log_{6} (\sqrt{6}) - \log_{6} (6)$

Step 8

Logarithm base $6$ of $6$ is $1$ .

$\log_{6} (\sqrt{6}) - 1 \cdot 1$

Step 9

Multiply $- 1$ by $1$ .

$\log_{6} (\sqrt{6}) - 1$

Step 10

Rewrite $\log_{6} (\sqrt{6})$ using the change of base formula.

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Step 10.1

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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} - 1$

Step 10.2

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

$\frac{\log (\sqrt{6})}{\log (6)} - 1$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\sqrt{6})}{\log (6)} - 1$

Decimal Form:

$- 0.5$