Precalculus Examples

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

Step 1

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

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

Step 2

Raise $2$ to the power of $5$ .

$\log_{2} (\sqrt{32})$

Step 3

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 3.1

Factor $16$ out of $32$ .

$\log_{2} (\sqrt{16 (2)})$

Step 3.2

Rewrite $16$ as $4^{2}$ .

$\log_{2} (\sqrt{4^{2} \cdot 2})$

Step 4

Pull terms out from under the radical.

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

Step 5

Rewrite $\log_{2} (4 \sqrt{2})$ using the change of base formula.

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Step 5.1

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 5.2

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

$\frac{\log (4 \sqrt{2})}{\log (2)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (4 \sqrt{2})}{\log (2)}$

Decimal Form:

$2.5$