Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify the denominator.

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Step 4.1

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

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

Step 4.2

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 $b = 1$ .

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

Step 5

Reduce the expression $\frac{(x - 1) x}{x (x + 1)}$ by cancelling the common factors.

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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

Step 6

Multiply the numerator by the reciprocal of the denominator.

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

Step 7

Cancel the common factor of $x - 1$ .

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Step 7.1

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

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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

Step 8

Multiply $\frac{1}{x + 1}$ by $\frac{1}{x + 1}$ .

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Add $1$ and $1$ .

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