Precalculus Examples

$\log_{5} (\sqrt{8})$

Step 1

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

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Step 1.1

Factor $4$ out of $8$ .

$\log_{5} (\sqrt{4 (2)})$

Step 1.2

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

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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Step 3.1

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 3.2

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

$\frac{\log (2 \sqrt{2})}{\log (5)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (2 \sqrt{2})}{\log (5)}$

Decimal Form:

$0.64601483 \dots$