Precalculus Examples

$\frac{1}{2} \cdot \ln (9) - 2 (\ln (x + 2) + \ln (x - 2))$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{2} \ln (9)$ by moving $\frac{1}{2}$ inside the logarithm.

$\ln (9^{\frac{1}{2}}) - 2 (\ln (x + 2) + \ln (x - 2))$

Step 1.2

Rewrite $9$ as $3^{2}$ .

$\ln ({(3^{2})}^{\frac{1}{2}}) - 2 (\ln (x + 2) + \ln (x - 2))$

Step 1.3

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\ln (3^{2 (\frac{1}{2})}) - 2 (\ln (x + 2) + \ln (x - 2))$

Step 1.4

Cancel the common factor of $2$ .

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Step 1.4.1

Cancel the common factor.

$\ln (3^{2 (\frac{1}{2})}) - 2 (\ln (x + 2) + \ln (x - 2))$

Step 1.4.2

Rewrite the expression.

$\ln (3^{1}) - 2 (\ln (x + 2) + \ln (x - 2))$

Step 1.5

Evaluate the exponent.

$\ln (3) - 2 (\ln (x + 2) + \ln (x - 2))$

Step 1.6

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

$\ln (3) - 2 \ln ((x + 2) (x - 2))$

Step 1.7

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

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Step 1.7.1

Apply the distributive property.

$\ln (3) - 2 \ln (x (x - 2) + 2 (x - 2))$

Step 1.7.2

Apply the distributive property.

$\ln (3) - 2 \ln (x \cdot x + x \cdot - 2 + 2 (x - 2))$

Step 1.7.3

Apply the distributive property.

$\ln (3) - 2 \ln (x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2)$

Step 1.8

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

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Step 1.8.1

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

$\ln (3) - 2 \ln (x \cdot x - 2 x + 2 x + 2 \cdot - 2)$

Step 1.8.2

Add $- 2 x$ and $2 x$ .

$\ln (3) - 2 \ln (x \cdot x + 0 + 2 \cdot - 2)$

Step 1.8.3

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

$\ln (3) - 2 \ln (x \cdot x + 2 \cdot - 2)$

Step 1.9

Simplify each term.

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Step 1.9.1

Multiply $x$ by $x$ .

$\ln (3) - 2 \ln (x^{2} + 2 \cdot - 2)$

Step 1.9.2

Multiply $2$ by $- 2$ .

$\ln (3) - 2 \ln (x^{2} - 4)$

Step 1.10

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

$\ln (3) - \ln ({(x^{2} - 4)}^{2})$

Step 2

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

$\ln (\frac{3}{{(x^{2} - 4)}^{2}})$

Step 3

Simplify the denominator.

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Step 3.1

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

$\ln (\frac{3}{{(x^{2} - 2^{2})}^{2}})$

Step 3.2

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 $b = 2$ .

$\ln (\frac{3}{{((x + 2) (x - 2))}^{2}})$

Step 3.3

Apply the product rule to $(x + 2) (x - 2)$ .

$\ln (\frac{3}{{(x + 2)}^{2} {(x - 2)}^{2}})$