Algebra Examples

$\ln (x) - (\ln (x + 1) + \ln (x - 1))$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

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

$\ln (x) - \ln ((x + 1) (x - 1))$

Step 1.2

Expand $(x + 1) (x - 1)$ using the FOIL Method.

Tap for more steps...

Step 1.2.1

Apply the distributive property.

$\ln (x) - \ln (x (x - 1) + 1 (x - 1))$

Step 1.2.2

Apply the distributive property.

$\ln (x) - \ln (x \cdot x + x \cdot - 1 + 1 (x - 1))$

Step 1.2.3

Apply the distributive property.

$\ln (x) - \ln (x \cdot x + x \cdot - 1 + 1 x + 1 \cdot - 1)$

Step 1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.3.1

Simplify each term.

Tap for more steps...

Step 1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.3.1.2

Move $- 1$ to the left of $x$ .

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

Step 1.3.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 1.3.1.4

Multiply $x$ by $1$ .

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

Step 1.3.1.5

Multiply $- 1$ by $1$ .

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

Step 1.3.2

Add $- x$ and $x$ .

$\ln (x) - \ln (x^{2} + 0 - 1)$

Step 1.3.3

Add $x^{2}$ and $0$ .

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

Step 2

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

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

Step 3

Simplify the denominator.

Tap for more steps...

Step 3.1

Rewrite $1$ as $1^{2}$ .

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

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 $b = 1$ .

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