Precalculus Examples

$\ln (\frac{x}{x - 10}) + \ln (\frac{x + 10}{x}) - \ln (x^{2} - 100)$

Step 1

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

$\ln (\frac{x}{x - 10} \cdot \frac{x + 10}{x}) - \ln (x^{2} - 100)$

Step 2

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

$\ln (\frac{\frac{x}{x - 10} \cdot \frac{x + 10}{x}}{x^{2} - 100})$

Step 3

Multiply $\frac{x}{x - 10}$ by $\frac{x + 10}{x}$ .

$\ln (\frac{\frac{x (x + 10)}{(x - 10) x}}{x^{2} - 100})$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $100$ as $10^{2}$ .

$\ln (\frac{\frac{x (x + 10)}{(x - 10) x}}{x^{2} - 10^{2}})$

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 $b = 10$ .

$\ln (\frac{\frac{x (x + 10)}{(x - 10) x}}{(x + 10) (x - 10)})$

Step 5

Reduce the expression $\frac{x (x + 10)}{(x - 10) x}$ by cancelling the common factors.

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Step 5.1

Cancel the common factor.

$\ln (\frac{\frac{x (x + 10)}{(x - 10) x}}{(x + 10) (x - 10)})$

Step 5.2

Rewrite the expression.

$\ln (\frac{\frac{x + 10}{x - 10}}{(x + 10) (x - 10)})$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{x + 10}{x - 10} \cdot \frac{1}{(x + 10) (x - 10)})$

Step 7

Cancel the common factor of $x + 10$ .

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Step 7.1

Cancel the common factor.

$\ln (\frac{x + 10}{x - 10} \cdot \frac{1}{(x + 10) (x - 10)})$

Step 7.2

Rewrite the expression.

$\ln (\frac{1}{x - 10} \cdot \frac{1}{x - 10})$

Step 8

Multiply $\frac{1}{x - 10}$ by $\frac{1}{x - 10}$ .

$\ln (\frac{1}{(x - 10) (x - 10)})$

Step 9

Raise $x - 10$ to the power of $1$ .

$\ln (\frac{1}{{(x - 10)}^{1} (x - 10)})$

Step 10

Raise $x - 10$ to the power of $1$ .

$\ln (\frac{1}{{(x - 10)}^{1} {(x - 10)}^{1}})$

Step 11

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

$\ln (\frac{1}{{(x - 10)}^{1 + 1}})$

Step 12

Add $1$ and $1$ .

$\ln (\frac{1}{{(x - 10)}^{2}})$