Precalculus Examples

$\log (x - 2) + \log (x + 2) - 0.5 \log (x^{2} + 4)$

Step 1

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

$\log ((x - 2) (x + 2)) - 0.5 \log (x^{2} + 4)$

Step 2

Simplify each term.

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Step 2.1

Expand $(x - 2) (x + 2)$ using the FOIL Method.

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Step 2.1.1

Apply the distributive property.

$\log (x (x + 2) - 2 (x + 2)) - 0.5 \log (x^{2} + 4)$

Step 2.1.2

Apply the distributive property.

$\log (x \cdot x + x \cdot 2 - 2 (x + 2)) - 0.5 \log (x^{2} + 4)$

Step 2.1.3

Apply the distributive property.

$\log (x \cdot x + x \cdot 2 - 2 x - 2 \cdot 2) - 0.5 \log (x^{2} + 4)$

Step 2.2

Combine the opposite terms in $x \cdot x + x \cdot 2 - 2 x - 2 \cdot 2$ .

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Step 2.2.1

Reorder the factors in the terms $x \cdot 2$ and $- 2 x$ .

$\log (x \cdot x + 2 x - 2 x - 2 \cdot 2) - 0.5 \log (x^{2} + 4)$

Step 2.2.2

Subtract $2 x$ from $2 x$ .

$\log (x \cdot x + 0 - 2 \cdot 2) - 0.5 \log (x^{2} + 4)$

Step 2.2.3

Add $x \cdot x$ and $0$ .

$\log (x \cdot x - 2 \cdot 2) - 0.5 \log (x^{2} + 4)$

Step 2.3

Simplify each term.

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Step 2.3.1

Multiply $x$ by $x$ .

$\log (x^{2} - 2 \cdot 2) - 0.5 \log (x^{2} + 4)$

Step 2.3.2

Multiply $- 2$ by $2$ .

$\log (x^{2} - 4) - 0.5 \log (x^{2} + 4)$

Step 2.4

Simplify $- 0.5 \log (x^{2} + 4)$ by moving $0.5$ inside the logarithm.

$\log (x^{2} - 4) - \log ({(x^{2} + 4)}^{0.5})$

Step 3

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

$\log (\frac{x^{2} - 4}{{(x^{2} + 4)}^{0.5}})$

Step 4

Simplify the numerator.

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Step 4.1

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

$\log (\frac{x^{2} - 2^{2}}{{(x^{2} + 4)}^{0.5}})$

Step 4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\log (\frac{(x + 2) (x - 2)}{{(x^{2} + 4)}^{0.5}})$