Precalculus Examples

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

Step 1

Rewrite $\sqrt{\frac{1}{25}}$ as $\frac{\sqrt{1}}{\sqrt{25}}$ .

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

Step 2

Any root of $1$ is $1$ .

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

Step 3

Simplify the denominator.

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Step 3.1

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

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

Step 3.2

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

$\log_{5} (4 (\frac{1}{5}))$

Step 4

Combine $4$ and $\frac{1}{5}$ .

$\log_{5} (\frac{4}{5})$

Step 5

Rewrite $\frac{4}{5}$ as $4 \cdot 5^{- 1}$ .

$\log_{5} (4 \cdot 5^{- 1})$

Step 6

Rewrite $\log_{5} (4 \cdot 5^{- 1})$ as $\log_{5} (4) + \log_{5} (5^{- 1})$ .

$\log_{5} (4) + \log_{5} (5^{- 1})$

Step 7

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

$\log_{5} (4) - \log_{5} (5)$

Step 8

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

$\log_{5} (4) - 1 \cdot 1$

Step 9

Multiply $- 1$ by $1$ .

$\log_{5} (4) - 1$

Step 10

Simplify each term.

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Step 10.1

Rewrite $\log_{5} (4)$ as $\log_{5} (2^{2})$ .

$\log_{5} (2^{2}) - 1$

Step 10.2

Expand $\log_{5} (2^{2})$ by moving $2$ outside the logarithm.

$2 \log_{5} (2) - 1$

Step 11

Simplify each term.

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Step 11.1

Rewrite $\log_{5} (2)$ using the change of base formula.

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Step 11.1.1

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 $0$ .

$2 \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} - 1$

Step 11.1.2

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

$2 \frac{\log (2)}{\log (5)} - 1$

Step 11.2

Multiply $2 \frac{\log (2)}{\log (5)}$ .

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Step 11.2.1

Combine $2$ and $\frac{\log (2)}{\log (5)}$ .

$\frac{2 \log (2)}{\log (5)} - 1$

Step 11.2.2

Simplify $2 \log (2)$ by moving $2$ inside the logarithm.

$\frac{\log (2^{2})}{\log (5)} - 1$

Step 11.3

Raise $2$ to the power of $2$ .

$\frac{\log (4)}{\log (5)} - 1$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (4)}{\log (5)} - 1$

Decimal Form:

$- 0.13864688 \dots$