Precalculus Examples

$\ln (x) - 4 (\ln (x + 2) + \ln (x - 2))$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

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

$\ln (x) - 4 \ln ((x + 2) (x - 2))$

Step 1.2

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

Tap for more steps...

Step 1.2.1

Apply the distributive property.

$\ln (x) - 4 \ln (x (x - 2) + 2 (x - 2))$

Step 1.2.2

Apply the distributive property.

$\ln (x) - 4 \ln (x \cdot x + x \cdot - 2 + 2 (x - 2))$

Step 1.2.3

Apply the distributive property.

$\ln (x) - 4 \ln (x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2)$

Step 1.3

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

Tap for more steps...

Step 1.3.1

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

$\ln (x) - 4 \ln (x \cdot x - 2 x + 2 x + 2 \cdot - 2)$

Step 1.3.2

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

$\ln (x) - 4 \ln (x \cdot x + 0 + 2 \cdot - 2)$

Step 1.3.3

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

$\ln (x) - 4 \ln (x \cdot x + 2 \cdot - 2)$

Step 1.4

Simplify each term.

Tap for more steps...

Step 1.4.1

Multiply $x$ by $x$ .

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

Step 1.4.2

Multiply $2$ by $- 2$ .

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

Step 1.5

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

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

Step 2

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

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

Step 3

Simplify the denominator.

Tap for more steps...

Step 3.1

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

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

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 $b = 2$ .

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

Step 3.3

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

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