Precalculus Examples

$\log_{4} (\sqrt{3 \cdot 2 \cdot 5})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 \cdot 2 \cdot 5}$ as ${(3 \cdot 2 \cdot 5)}^{\frac{1}{2}}$ .

$\log_{4} ({(3 \cdot 2 \cdot 5)}^{\frac{1}{2}})$

Step 2

Expand $\log_{4} ({(3 \cdot 2 \cdot 5)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{4} (3 \cdot 2 \cdot 5)$

Step 3

Multiply $3$ by $2$ .

$\frac{1}{2} \log_{4} (6 \cdot 5)$

Step 4

Multiply $6$ by $5$ .

$\frac{1}{2} \log_{4} (30)$

Step 5

Rewrite $\log_{4} (30)$ as $\log_{4} (2 \cdot 3 \cdot 5)$ .

$\frac{1}{2} \log_{4} (2 \cdot 3 \cdot 5)$

Step 6

Rewrite $\log_{4} (2 \cdot 3 \cdot 5)$ as $\log_{4} (2 \cdot 3) + \log_{4} (5)$ .

$\frac{1}{2} (\log_{4} (2 \cdot 3) + \log_{4} (5))$

Step 7

Multiply $2$ by $3$ .

$\frac{1}{2} (\log_{4} (6) + \log_{4} (5))$

Step 8

Simplify each term.

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Step 8.1

Rewrite $\log_{4} (6)$ as $\log_{4} (2 (3))$ .

$\frac{1}{2} (\log_{4} (2 (3)) + \log_{4} (5))$

Step 8.2

Rewrite $\log_{4} (2 (3))$ as $\log_{4} (2) + \log_{4} (3)$ .

$\frac{1}{2} (\log_{4} (2) + \log_{4} (3) + \log_{4} (5))$

Step 8.3

Logarithm base $4$ of $2$ is $\frac{1}{2}$ .

$\frac{1}{2} (\frac{1}{2} + \log_{4} (3) + \log_{4} (5))$

Step 9

Apply the distributive property.

$\frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \log_{4} (3) + \frac{1}{2} \log_{4} (5)$

Step 10

Simplify.

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Step 10.1

Multiply $\frac{1}{2} \cdot \frac{1}{2}$ .

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Step 10.1.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$\frac{1}{2 \cdot 2} + \frac{1}{2} \log_{4} (3) + \frac{1}{2} \log_{4} (5)$

Step 10.1.2

Multiply $2$ by $2$ .

$\frac{1}{4} + \frac{1}{2} \log_{4} (3) + \frac{1}{2} \log_{4} (5)$

Step 10.2

Combine $\frac{1}{2}$ and $\log_{4} (3)$ .

$\frac{1}{4} + \frac{\log_{4} (3)}{2} + \frac{1}{2} \log_{4} (5)$

Step 10.3

Combine $\frac{1}{2}$ and $\log_{4} (5)$ .

$\frac{1}{4} + \frac{\log_{4} (3)}{2} + \frac{\log_{4} (5)}{2}$