Precalculus Examples

$\log_{6} (\sqrt[3]{1296})$

Step 1

Rewrite $1296$ as $6^{3} \cdot 6$ .

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Step 1.1

Factor $216$ out of $1296$ .

$\log_{6} (\sqrt[3]{216 (6)})$

Step 1.2

Rewrite $216$ as $6^{3}$ .

$\log_{6} (\sqrt[3]{6^{3} \cdot 6})$

Step 2

Pull terms out from under the radical.

$\log_{6} (6 \sqrt[3]{6})$

Step 3

Rewrite $\log_{6} (6 \sqrt[3]{6})$ 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 $b = 10$ .

$\frac{\log (6 \sqrt[3]{6})}{\log (6)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (6 \sqrt[3]{6})}{\log (6)}$

Decimal Form:

$1.\overline{3}$