Precalculus Examples

$\log_{4} (\sqrt{12}) + \log_{4} (\sqrt{9}) - \log_{4} (\sqrt{27})$

Step 1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{4} (\sqrt{12} \sqrt{9}) - \log_{4} (\sqrt{27})$

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{4} (\frac{\sqrt{12} \sqrt{9}}{\sqrt{27}})$

Step 3

Combine $\sqrt{12}$ and $\sqrt{27}$ into a single radical.

$\log_{4} (\sqrt{\frac{12}{27}} \sqrt{9})$

Step 4

Cancel the common factor of $12$ and $27$ .

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Step 4.1

Factor $3$ out of $12$ .

$\log_{4} (\sqrt{\frac{3 (4)}{27}} \sqrt{9})$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $3$ out of $27$ .

$\log_{4} (\sqrt{\frac{3 \cdot 4}{3 \cdot 9}} \sqrt{9})$

Step 4.2.2

Cancel the common factor.

$\log_{4} (\sqrt{\frac{3 \cdot 4}{3 \cdot 9}} \sqrt{9})$

Step 4.2.3

Rewrite the expression.

$\log_{4} (\sqrt{\frac{4}{9}} \sqrt{9})$

Step 5

Rewrite $\sqrt{\frac{4}{9}}$ as $\frac{\sqrt{4}}{\sqrt{9}}$ .

$\log_{4} (\frac{\sqrt{4}}{\sqrt{9}} \sqrt{9})$

Step 6

Cancel the common factor of $\sqrt{9}$ .

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Step 6.1

Cancel the common factor.

$\log_{4} (\frac{\sqrt{4}}{\sqrt{9}} \sqrt{9})$

Step 6.2

Rewrite the expression.

$\log_{4} (\sqrt{4})$

Step 7

Rewrite $4$ as $2^{2}$ .

$\log_{4} (\sqrt{2^{2}})$

Step 8

Pull terms out from under the radical, assuming positive real numbers.

$\log_{4} (2)$

Step 9

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

$\frac{1}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$