Precalculus Examples

$\log_{4} (\frac{\sqrt{12}}{11})$

Step 1

Rewrite $\log_{4} (\frac{\sqrt{12}}{11})$ as $\log_{4} (\sqrt{12}) - \log_{4} (11)$ .

$\log_{4} (\sqrt{12}) - \log_{4} (11)$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{12}$ as $12^{\frac{1}{2}}$ .

$\log_{4} (12^{\frac{1}{2}}) - \log_{4} (11)$

Step 3

Expand $\log_{4} (12^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{4} (12) - \log_{4} (11)$

Step 4

Rewrite $\log_{4} (12)$ as $\log_{4} (2^{2} \cdot 3)$ .

$\frac{1}{2} \log_{4} (2^{2} \cdot 3) - \log_{4} (11)$

Step 5

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

$\frac{1}{2} (\log_{4} (2^{2}) + \log_{4} (3)) - \log_{4} (11)$

Step 6

Simplify each term.

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Step 6.1

Expand $\log_{4} (2^{2})$ by moving $2$ outside the logarithm.

$\frac{1}{2} (2 \log_{4} (2) + \log_{4} (3)) - \log_{4} (11)$

Step 6.2

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

$\frac{1}{2} (2 (\frac{1}{2}) + \log_{4} (3)) - \log_{4} (11)$

Step 6.3

Cancel the common factor of $2$ .

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Step 6.3.1

Cancel the common factor.

$\frac{1}{2} (2 (\frac{1}{2}) + \log_{4} (3)) - \log_{4} (11)$

Step 6.3.2

Rewrite the expression.

$\frac{1}{2} (1 + \log_{4} (3)) - \log_{4} (11)$

Step 7

Apply the distributive property.

$\frac{1}{2} \cdot 1 + \frac{1}{2} \log_{4} (3) - \log_{4} (11)$

Step 8

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

$\frac{1}{2} + \frac{1}{2} \log_{4} (3) - \log_{4} (11)$

Step 9

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

$\frac{1}{2} + \frac{\log_{4} (3)}{2} - \log_{4} (11)$