Precalculus Examples

$\log_{9} (3 \sqrt[3]{11 \cdot 8})$

Step 1

Rewrite $\log_{9} (3 \sqrt[3]{11 \cdot 8})$ as $\log_{9} (3) + \log_{9} (\sqrt[3]{11 \cdot 8})$ .

$\log_{9} (3) + \log_{9} (\sqrt[3]{11 \cdot 8})$

Step 2

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

$\log_{9} (3) + \log_{9} ({(11 \cdot 8)}^{\frac{1}{3}})$

Step 3

Expand $\log_{9} ({(11 \cdot 8)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\log_{9} (3) + \frac{1}{3} \log_{9} (11 \cdot 8)$

Step 4

Logarithm base $9$ of $3$ is $\frac{1}{2}$ .

$\frac{1}{2} + \frac{1}{3} \log_{9} (11 \cdot 8)$

Step 5

Multiply $11$ by $8$ .

$\frac{1}{2} + \frac{1}{3} \log_{9} (88)$

Step 6

Rewrite $\log_{9} (88)$ as $\log_{9} (2^{3} \cdot 11)$ .

$\frac{1}{2} + \frac{1}{3} \log_{9} (2^{3} \cdot 11)$

Step 7

Rewrite $\log_{9} (2^{3} \cdot 11)$ as $\log_{9} (2^{3}) + \log_{9} (11)$ .

$\frac{1}{2} + \frac{1}{3} (\log_{9} (2^{3}) + \log_{9} (11))$

Step 8

Expand $\log_{9} (2^{3})$ by moving $3$ outside the logarithm.

$\frac{1}{2} + \frac{1}{3} (3 \log_{9} (2) + \log_{9} (11))$

Step 9

Apply the distributive property.

$\frac{1}{2} + \frac{1}{3} (3 \log_{9} (2)) + \frac{1}{3} \log_{9} (11)$

Step 10

Cancel the common factor of $3$ .

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Step 10.1

Factor $3$ out of $3 \log_{9} (2)$ .

$\frac{1}{2} + \frac{1}{3} (3 (\log_{9} (2))) + \frac{1}{3} \log_{9} (11)$

Step 10.2

Cancel the common factor.

$\frac{1}{2} + \frac{1}{3} (3 \log_{9} (2)) + \frac{1}{3} \log_{9} (11)$

Step 10.3

Rewrite the expression.

$\frac{1}{2} + \log_{9} (2) + \frac{1}{3} \log_{9} (11)$

Step 11

Combine $\frac{1}{3}$ and $\log_{9} (11)$ .

$\frac{1}{2} + \log_{9} (2) + \frac{\log_{9} (11)}{3}$