Precalculus Examples

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

Step 1

Simplify $2 \log (x - 1)$ by moving $2$ inside the logarithm.

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

Step 2

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

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

Step 3

Simplify the denominator.

Tap for more steps...

Step 3.1

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

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

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 $b = 1$ .

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

Step 4

Cancel the common factor of ${(x - 1)}^{2}$ and $x - 1$ .

Tap for more steps...

Step 4.1

Factor $x - 1$ out of ${(x - 1)}^{2}$ .

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

Step 4.2

Cancel the common factors.

Tap for more steps...

Step 4.2.1

Factor $x - 1$ out of $(x + 1) (x - 1)$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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