Precalculus Examples

$\log_{9} (\sqrt{5 \cdot 7 \cdot 8})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 \cdot 7 \cdot 8}$ as ${(5 \cdot 7 \cdot 8)}^{\frac{1}{2}}$ .

$\log_{9} ({(5 \cdot 7 \cdot 8)}^{\frac{1}{2}})$

Step 2

Expand $\log_{9} ({(5 \cdot 7 \cdot 8)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{9} (5 \cdot 7 \cdot 8)$

Step 3

Multiply $5$ by $7$ .

$\frac{1}{2} \log_{9} (35 \cdot 8)$

Step 4

Multiply $35$ by $8$ .

$\frac{1}{2} \log_{9} (280)$

Step 5

Rewrite $\log_{9} (280)$ as $\log_{9} (2^{3} \cdot 5 \cdot 7)$ .

$\frac{1}{2} \log_{9} (2^{3} \cdot 5 \cdot 7)$

Step 6

Rewrite $\log_{9} (2^{3} \cdot 5 \cdot 7)$ as $\log_{9} (2^{3} \cdot 5) + \log_{9} (7)$ .

$\frac{1}{2} (\log_{9} (2^{3} \cdot 5) + \log_{9} (7))$

Step 7

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

$\frac{1}{2} (\log_{9} (8 \cdot 5) + \log_{9} (7))$

Step 8

Multiply $8$ by $5$ .

$\frac{1}{2} (\log_{9} (40) + \log_{9} (7))$

Step 9

Simplify each term.

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Step 9.1

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

$\frac{1}{2} (\log_{9} (2^{3} \cdot 5) + \log_{9} (7))$

Step 9.2

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

$\frac{1}{2} (\log_{9} (2^{3}) + \log_{9} (5) + \log_{9} (7))$

Step 9.3

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

$\frac{1}{2} (3 \log_{9} (2) + \log_{9} (5) + \log_{9} (7))$

Step 10

Apply the distributive property.

$\frac{1}{2} (3 \log_{9} (2)) + \frac{1}{2} \log_{9} (5) + \frac{1}{2} \log_{9} (7)$

Step 11

Simplify.

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Step 11.1

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

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Step 11.1.1

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

$\frac{3}{2} \log_{9} (2) + \frac{1}{2} \log_{9} (5) + \frac{1}{2} \log_{9} (7)$

Step 11.1.2

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

$\frac{3 \log_{9} (2)}{2} + \frac{1}{2} \log_{9} (5) + \frac{1}{2} \log_{9} (7)$

Step 11.2

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

$\frac{3 \log_{9} (2)}{2} + \frac{\log_{9} (5)}{2} + \frac{1}{2} \log_{9} (7)$

Step 11.3

Combine $\frac{1}{2}$ and $\log_{9} (7)$ .

$\frac{3 \log_{9} (2)}{2} + \frac{\log_{9} (5)}{2} + \frac{\log_{9} (7)}{2}$