Precalculus Examples

$\log_{216} (\sqrt{36 \sqrt{6}})$

Step 1

Rewrite $36$ as $6^{2}$ .

$\log_{216} (\sqrt{6^{2} \sqrt{6}})$

Step 2

Pull terms out from under the radical.

$\log_{216} (6 \sqrt{\sqrt{6}})$

Step 3

Rewrite $\sqrt{\sqrt{6}}$ as $\sqrt[4]{6}$ .

$\log_{216} (6 \sqrt[4]{6})$

Step 4

Rewrite $\log_{216} (6 \sqrt[4]{6})$ using the change of base formula.

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Step 4.1

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 4.2

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

$\frac{\log (6 \sqrt[4]{6})}{\log (216)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (6 \sqrt[4]{6})}{\log (216)}$

Decimal Form:

$0.41\overline{6}$