Precalculus Examples

$\log_{9} (3 \sqrt{11 \cdot 8 \cdot 5})$

Step 1

Rewrite $\log_{9} (3 \sqrt{11 \cdot 8 \cdot 5})$ as $\log_{9} (3) + \log_{9} (\sqrt{11 \cdot 8 \cdot 5})$ .

$\log_{9} (3) + \log_{9} (\sqrt{11 \cdot 8 \cdot 5})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11 \cdot 8 \cdot 5}$ as ${(11 \cdot 8 \cdot 5)}^{\frac{1}{2}}$ .

$\log_{9} (3) + \log_{9} ({(11 \cdot 8 \cdot 5)}^{\frac{1}{2}})$

Step 3

Expand $\log_{9} ({(11 \cdot 8 \cdot 5)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{9} (3) + \frac{1}{2} \log_{9} (11 \cdot 8 \cdot 5)$

Step 4

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

$\frac{1}{2} + \frac{1}{2} \log_{9} (11 \cdot 8 \cdot 5)$

Step 5

Multiply $11$ by $8$ .

$\frac{1}{2} + \frac{1}{2} \log_{9} (88 \cdot 5)$

Step 6

Multiply $88$ by $5$ .

$\frac{1}{2} + \frac{1}{2} \log_{9} (440)$

Step 7

Rewrite $\log_{9} (440)$ as $\log_{9} (2^{3} \cdot 5 \cdot 11)$ .

$\frac{1}{2} + \frac{1}{2} \log_{9} (2^{3} \cdot 5 \cdot 11)$

Step 8

Rewrite $\log_{9} (2^{3} \cdot 5 \cdot 11)$ as $\log_{9} (2^{3} \cdot 5) + \log_{9} (11)$ .

$\frac{1}{2} + \frac{1}{2} (\log_{9} (2^{3} \cdot 5) + \log_{9} (11))$

Step 9

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

$\frac{1}{2} + \frac{1}{2} (\log_{9} (8 \cdot 5) + \log_{9} (11))$

Step 10

Multiply $8$ by $5$ .

$\frac{1}{2} + \frac{1}{2} (\log_{9} (40) + \log_{9} (11))$

Step 11

Simplify each term.

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Step 11.1

Rewrite $\log_{9} (40)$ as $\log_{9} (2^{3} \cdot 5)$ .

$\frac{1}{2} + \frac{1}{2} (\log_{9} (2^{3} \cdot 5) + \log_{9} (11))$

Step 11.2

Rewrite $\log_{9} (2^{3} \cdot 5)$ as $\log_{9} (2^{3}) + \log_{9} (5)$ .

$\frac{1}{2} + \frac{1}{2} (\log_{9} (2^{3}) + \log_{9} (5) + \log_{9} (11))$

Step 11.3

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

$\frac{1}{2} + \frac{1}{2} (3 \log_{9} (2) + \log_{9} (5) + \log_{9} (11))$

Step 12

Apply the distributive property.

$\frac{1}{2} + \frac{1}{2} (3 \log_{9} (2)) + \frac{1}{2} \log_{9} (5) + \frac{1}{2} \log_{9} (11)$

Step 13

Simplify.

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Step 13.1

Multiply $\frac{1}{2} (3 \log_{9} (2))$ .

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Step 13.1.1

Combine $3$ and $\frac{1}{2}$ .

$\frac{1}{2} + \frac{3}{2} \log_{9} (2) + \frac{1}{2} \log_{9} (5) + \frac{1}{2} \log_{9} (11)$

Step 13.1.2

Combine $\frac{3}{2}$ and $\log_{9} (2)$ .

$\frac{1}{2} + \frac{3 \log_{9} (2)}{2} + \frac{1}{2} \log_{9} (5) + \frac{1}{2} \log_{9} (11)$

Step 13.2

Combine $\frac{1}{2}$ and $\log_{9} (5)$ .

$\frac{1}{2} + \frac{3 \log_{9} (2)}{2} + \frac{\log_{9} (5)}{2} + \frac{1}{2} \log_{9} (11)$

Step 13.3

Combine $\frac{1}{2}$ and $\log_{9} (11)$ .

$\frac{1}{2} + \frac{3 \log_{9} (2)}{2} + \frac{\log_{9} (5)}{2} + \frac{\log_{9} (11)}{2}$