Write as a Single Logarithm natural log of x/(x-7)+ natural log of (x+7)/x- natural log of x^2-49

$\ln (\frac{x}{x - 7}) + \ln (\frac{x + 7}{x}) - \ln (x^{2} - 49)$

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

$\ln (\frac{x}{x - 7} \cdot \frac{x + 7}{x}) - \ln (x^{2} - 49)$

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

$\ln (\frac{\frac{x}{x - 7} \cdot \frac{x + 7}{x}}{x^{2} - 49})$

Multiply $\frac{x}{x - 7}$ by $\frac{x + 7}{x}$ .

$\ln (\frac{\frac{x (x + 7)}{(x - 7) x}}{x^{2} - 49})$

Simplify the denominator.

Rewrite $49$ as $7^{2}$ .

$\ln (\frac{\frac{x (x + 7)}{(x - 7) x}}{x^{2} - 7^{2}})$

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 $b = 7$ .

$\ln (\frac{\frac{x (x + 7)}{(x - 7) x}}{(x + 7) (x - 7)})$

Reduce the expression $\frac{x (x + 7)}{(x - 7) x}$ by cancelling the common factors.

Cancel the common factor.

$\ln (\frac{\frac{x (x + 7)}{(x - 7) x}}{(x + 7) (x - 7)})$

Rewrite the expression.

$\ln (\frac{\frac{x + 7}{x - 7}}{(x + 7) (x - 7)})$

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{x + 7}{x - 7} \cdot \frac{1}{(x + 7) (x - 7)})$

Cancel the common factor of $x + 7$ .

Cancel the common factor.

$\ln (\frac{x + 7}{x - 7} \cdot \frac{1}{(x + 7) (x - 7)})$

Rewrite the expression.

$\ln (\frac{1}{x - 7} \cdot \frac{1}{x - 7})$

Multiply $\frac{1}{x - 7}$ by $\frac{1}{x - 7}$ .

$\ln (\frac{1}{(x - 7) (x - 7)})$

Raise $x - 7$ to the power of $1$ .

$\ln (\frac{1}{{(x - 7)}^{1} (x - 7)})$

Raise $x - 7$ to the power of $1$ .

$\ln (\frac{1}{{(x - 7)}^{1} {(x - 7)}^{1}})$

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

$\ln (\frac{1}{{(x - 7)}^{1 + 1}})$

Add $1$ and $1$ .

$\ln (\frac{1}{{(x - 7)}^{2}})$