Simplify/Condense natural log of x/(x-1)+ natural log of (x+1)/x- natural log of x^2-1

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

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

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

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

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

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

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

Simplify the denominator.

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

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

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 $b = 1$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $x + 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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