Precalculus Examples

$\ln (\frac{x}{x - 4}) + \ln (\frac{x + 4}{x}) - \ln (x^{2} - 16)$

Step 1

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

$\ln (\frac{x}{x - 4} \cdot \frac{x + 4}{x}) - \ln (x^{2} - 16)$

Step 2

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

$\ln (\frac{\frac{x}{x - 4} \cdot \frac{x + 4}{x}}{x^{2} - 16})$

Step 3

Multiply $\frac{x}{x - 4}$ by $\frac{x + 4}{x}$ .

$\ln (\frac{\frac{x (x + 4)}{(x - 4) x}}{x^{2} - 16})$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $16$ as $4^{2}$ .

$\ln (\frac{\frac{x (x + 4)}{(x - 4) x}}{x^{2} - 4^{2}})$

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 $b = 4$ .

$\ln (\frac{\frac{x (x + 4)}{(x - 4) x}}{(x + 4) (x - 4)})$

Step 5

Reduce the expression $\frac{x (x + 4)}{(x - 4) x}$ by cancelling the common factors.

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Step 5.1

Cancel the common factor.

$\ln (\frac{\frac{x (x + 4)}{(x - 4) x}}{(x + 4) (x - 4)})$

Step 5.2

Rewrite the expression.

$\ln (\frac{\frac{x + 4}{x - 4}}{(x + 4) (x - 4)})$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{x + 4}{x - 4} \cdot \frac{1}{(x + 4) (x - 4)})$

Step 7

Cancel the common factor of $x + 4$ .

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Step 7.1

Cancel the common factor.

$\ln (\frac{x + 4}{x - 4} \cdot \frac{1}{(x + 4) (x - 4)})$

Step 7.2

Rewrite the expression.

$\ln (\frac{1}{x - 4} \cdot \frac{1}{x - 4})$

Step 8

Multiply $\frac{1}{x - 4}$ by $\frac{1}{x - 4}$ .

$\ln (\frac{1}{(x - 4) (x - 4)})$

Step 9

Raise $x - 4$ to the power of $1$ .

$\ln (\frac{1}{{(x - 4)}^{1} (x - 4)})$

Step 10

Raise $x - 4$ to the power of $1$ .

$\ln (\frac{1}{{(x - 4)}^{1} {(x - 4)}^{1}})$

Step 11

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

$\ln (\frac{1}{{(x - 4)}^{1 + 1}})$

Step 12

Add $1$ and $1$ .

$\ln (\frac{1}{{(x - 4)}^{2}})$