Precalculus Examples

$\log_{5} (\sqrt{\frac{x}{25}})$

Step 1

Rewrite $\frac{x}{25}$ as ${(\frac{1}{5})}^{2} x$ .

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Step 1.1

Factor the perfect power $1^{2}$ out of $x$ .

$\log_{5} (\sqrt{\frac{1^{2} x}{25}})$

Step 1.2

Factor the perfect power $5^{2}$ out of $25$ .

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

Step 1.3

Rearrange the fraction $\frac{1^{2} x}{5^{2} \cdot 1}$ .

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

Step 2

Pull terms out from under the radical.

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

Step 3

Combine $\frac{1}{5}$ and $\sqrt{x}$ .

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Multiply $- 1$ by $1$ .

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

Step 9

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

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Step 9.1

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 9.2

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

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