Precalculus Examples

$\log_{5} (\sqrt[4]{\frac{1}{25}})$

Step 1

Rewrite $1$ as $1^{2}$ .

$\log_{5} (\sqrt[4]{\frac{1^{2}}{25}})$

Step 2

Rewrite $25$ as $5^{2}$ .

$\log_{5} (\sqrt[4]{\frac{1^{2}}{5^{2}}})$

Step 3

Rewrite $\frac{1^{2}}{5^{2}}$ as ${(\frac{1}{5})}^{2}$ .

$\log_{5} (\sqrt[4]{{(\frac{1}{5})}^{2}})$

Step 4

Rewrite $\sqrt[4]{{(\frac{1}{5})}^{2}}$ as $\sqrt{\sqrt{{(\frac{1}{5})}^{2}}}$ .

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

Step 5

Pull terms out from under the radical, assuming positive real numbers.

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

Step 6

Rewrite $\sqrt{\frac{1}{5}}$ as $\frac{\sqrt{1}}{\sqrt{5}}$ .

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

Step 7

Any root of $1$ is $1$ .

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

Step 8

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 9

Combine and simplify the denominator.

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Step 9.1

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 9.2

Raise $\sqrt{5}$ to the power of $1$ .

$\log_{5} (\frac{\sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}})$

Step 9.3

Raise $\sqrt{5}$ to the power of $1$ .

$\log_{5} (\frac{\sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}})$

Step 9.4

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

$\log_{5} (\frac{\sqrt{5}}{{\sqrt{5}}^{1 + 1}})$

Step 9.5

Add $1$ and $1$ .

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

Step 9.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 9.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

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

Step 9.6.2

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

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

Step 9.6.3

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

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

Step 9.6.4

Cancel the common factor of $2$ .

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Step 9.6.4.1

Cancel the common factor.

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

Step 9.6.4.2

Rewrite the expression.

$\log_{5} (\frac{\sqrt{5}}{5^{1}})$

Step 9.6.5

Evaluate the exponent.

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

Step 10

Rewrite $\frac{\sqrt{5}}{5}$ as $\sqrt{5} \cdot 5^{- 1}$ .

$\log_{5} (\sqrt{5} \cdot 5^{- 1})$

Step 11

Rewrite $\log_{5} (\sqrt{5} \cdot 5^{- 1})$ as $\log_{5} (\sqrt{5}) + \log_{5} (5^{- 1})$ .

$\log_{5} (\sqrt{5}) + \log_{5} (5^{- 1})$

Step 12

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

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

Step 13

Logarithm base $5$ of $5$ is $1$ .

$\log_{5} (\sqrt{5}) - 1 \cdot 1$

Step 14

Multiply $- 1$ by $1$ .

$\log_{5} (\sqrt{5}) - 1$

Step 15

Rewrite $\log_{5} (\sqrt{5})$ using the change of base formula.

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Step 15.1

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 15.2

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

$\frac{\log (\sqrt{5})}{\log (5)} - 1$

Step 16

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\sqrt{5})}{\log (5)} - 1$

Decimal Form:

$- 0.5$