Precalculus Examples

$\log_{3} (\sqrt{22} - 4) + \log_{3} (\sqrt{22} + 4)$

Step 1

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

$\log_{3} ((\sqrt{22} - 4) (\sqrt{22} + 4))$

Step 2

Expand $(\sqrt{22} - 4) (\sqrt{22} + 4)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\log_{3} (\sqrt{22} (\sqrt{22} + 4) - 4 (\sqrt{22} + 4))$

Step 2.2

Apply the distributive property.

$\log_{3} (\sqrt{22} \sqrt{22} + \sqrt{22} \cdot 4 - 4 (\sqrt{22} + 4))$

Step 2.3

Apply the distributive property.

$\log_{3} (\sqrt{22} \sqrt{22} + \sqrt{22} \cdot 4 - 4 \sqrt{22} - 4 \cdot 4)$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Combine using the product rule for radicals.

$\log_{3} (\sqrt{22 \cdot 22} + \sqrt{22} \cdot 4 - 4 \sqrt{22} - 4 \cdot 4)$

Step 3.1.2

Multiply $22$ by $22$ .

$\log_{3} (\sqrt{484} + \sqrt{22} \cdot 4 - 4 \sqrt{22} - 4 \cdot 4)$

Step 3.1.3

Rewrite $484$ as $22^{2}$ .

$\log_{3} (\sqrt{22^{2}} + \sqrt{22} \cdot 4 - 4 \sqrt{22} - 4 \cdot 4)$

Step 3.1.4

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

$\log_{3} (22 + \sqrt{22} \cdot 4 - 4 \sqrt{22} - 4 \cdot 4)$

Step 3.1.5

Move $4$ to the left of $\sqrt{22}$ .

$\log_{3} (22 + 4 \cdot \sqrt{22} - 4 \sqrt{22} - 4 \cdot 4)$

Step 3.1.6

Multiply $- 4$ by $4$ .

$\log_{3} (22 + 4 \sqrt{22} - 4 \sqrt{22} - 16)$

Step 3.2

Subtract $16$ from $22$ .

$\log_{3} (6 + 4 \sqrt{22} - 4 \sqrt{22})$

Step 3.3

Subtract $4 \sqrt{22}$ from $4 \sqrt{22}$ .

$\log_{3} (6 + 0)$

Step 3.4

Add $6$ and $0$ .

$\log_{3} (6)$

Step 4

Rewrite $\log_{3} (6)$ using the change of base formula.

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Step 4.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 4.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (6)}{\log (3)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (6)}{\log (3)}$

Decimal Form:

$1.63092975 \dots$