Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify each term.

Tap for more steps...

Step 3.1

Simplify the numerator.

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

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 $b = 1$ .

$\log (\frac{x (x + 1) (x - 1)}{x + 1}) - 2 \log (x)$

Step 3.2

Cancel the common factor of $x + 1$ .

Tap for more steps...

Step 3.2.1

Cancel the common factor.

$\log (\frac{x (x + 1) (x - 1)}{x + 1}) - 2 \log (x)$

Step 3.2.2

Divide $x (x - 1)$ by $1$ .

$\log (x (x - 1)) - 2 \log (x)$

Step 3.3

Apply the distributive property.

$\log (x \cdot x + x \cdot - 1) - 2 \log (x)$

Step 3.4

Multiply $x$ by $x$ .

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

Step 3.5

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

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

Step 3.6

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

$\log (x^{2} - x) - 2 \log (x)$

Step 3.7

Simplify $- 2 \log (x)$ by moving $2$ inside the logarithm.

$\log (x^{2} - x) - \log (x^{2})$

Step 4

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

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

Step 5

Factor $x$ out of $x^{2} - x$ .

Tap for more steps...

Step 5.1

Factor $x$ out of $x^{2}$ .

$\log (\frac{x \cdot x - x}{x^{2}})$

Step 5.2

Factor $x$ out of $- x$ .

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

Step 5.3

Factor $x$ out of $x \cdot x + x \cdot - 1$ .

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

Step 6

Cancel the common factors.

Tap for more steps...

Step 6.1

Factor $x$ out of $x^{2}$ .

$\log (\frac{x (x - 1)}{x \cdot x})$

Step 6.2

Cancel the common factor.

$\log (\frac{x (x - 1)}{x \cdot x})$

Step 6.3

Rewrite the expression.

$\log (\frac{x - 1}{x})$