Precalculus Examples

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

Step 1

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

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

Simplify the denominator.

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Step 3.1

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

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

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 $b = 1$ .

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

Step 4

Multiply $\frac{x}{x - 1} \cdot \frac{1}{(x + 1) (x - 1)}$ .

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Step 4.1

Multiply $\frac{x}{x - 1}$ by $\frac{1}{(x + 1) (x - 1)}$ .

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

Step 4.2

Raise $x - 1$ to the power of $1$ .

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

Step 4.3

Raise $x - 1$ to the power of $1$ .

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

Step 4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.5

Add $1$ and $1$ .

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