Precalculus Examples

$\log_{3} (\sqrt[3]{8 \cdot 7 \cdot 11})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{8 \cdot 7 \cdot 11}$ as ${(8 \cdot 7 \cdot 11)}^{\frac{1}{3}}$ .

$\log_{3} ({(8 \cdot 7 \cdot 11)}^{\frac{1}{3}})$

Step 2

Expand $\log_{3} ({(8 \cdot 7 \cdot 11)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log_{3} (8 \cdot 7 \cdot 11)$

Step 3

Multiply $8$ by $7$ .

$\frac{1}{3} \log_{3} (56 \cdot 11)$

Step 4

Multiply $56$ by $11$ .

$\frac{1}{3} \log_{3} (616)$

Step 5

Rewrite $\log_{3} (616)$ as $\log_{3} (2^{3} \cdot 7 \cdot 11)$ .

$\frac{1}{3} \log_{3} (2^{3} \cdot 7 \cdot 11)$

Step 6

Rewrite $\log_{3} (2^{3} \cdot 7 \cdot 11)$ as $\log_{3} (2^{3} \cdot 7) + \log_{3} (11)$ .

$\frac{1}{3} (\log_{3} (2^{3} \cdot 7) + \log_{3} (11))$

Step 7

Raise $2$ to the power of $3$ .

$\frac{1}{3} (\log_{3} (8 \cdot 7) + \log_{3} (11))$

Step 8

Multiply $8$ by $7$ .

$\frac{1}{3} (\log_{3} (56) + \log_{3} (11))$

Step 9

Simplify each term.

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Step 9.1

Rewrite $\log_{3} (56)$ as $\log_{3} (2^{3} \cdot 7)$ .

$\frac{1}{3} (\log_{3} (2^{3} \cdot 7) + \log_{3} (11))$

Step 9.2

Rewrite $\log_{3} (2^{3} \cdot 7)$ as $\log_{3} (2^{3}) + \log_{3} (7)$ .

$\frac{1}{3} (\log_{3} (2^{3}) + \log_{3} (7) + \log_{3} (11))$

Step 9.3

Expand $\log_{3} (2^{3})$ by moving $3$ outside the logarithm.

$\frac{1}{3} (3 \log_{3} (2) + \log_{3} (7) + \log_{3} (11))$

Step 10

Apply the distributive property.

$\frac{1}{3} (3 \log_{3} (2)) + \frac{1}{3} \log_{3} (7) + \frac{1}{3} \log_{3} (11)$

Step 11

Simplify.

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Step 11.1

Cancel the common factor of $3$ .

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Step 11.1.1

Factor $3$ out of $3 \log_{3} (2)$ .

$\frac{1}{3} (3 (\log_{3} (2))) + \frac{1}{3} \log_{3} (7) + \frac{1}{3} \log_{3} (11)$

Step 11.1.2

Cancel the common factor.

$\frac{1}{3} (3 \log_{3} (2)) + \frac{1}{3} \log_{3} (7) + \frac{1}{3} \log_{3} (11)$

Step 11.1.3

Rewrite the expression.

$\log_{3} (2) + \frac{1}{3} \log_{3} (7) + \frac{1}{3} \log_{3} (11)$

Step 11.2

Combine $\frac{1}{3}$ and $\log_{3} (7)$ .

$\log_{3} (2) + \frac{\log_{3} (7)}{3} + \frac{1}{3} \log_{3} (11)$

Step 11.3

Combine $\frac{1}{3}$ and $\log_{3} (11)$ .

$\log_{3} (2) + \frac{\log_{3} (7)}{3} + \frac{\log_{3} (11)}{3}$