Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify the numerator.

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Step 3.1

To write $x$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 3.3

Simplify the numerator.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

Add $1$ and $1$ .

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

Step 3.3.5

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

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

Step 3.3.6

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 $b = 1$ .

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

Step 4

Combine $\frac{(x + 1) (x - 1)}{x}$ and $x$ .

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

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

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

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Step 5.1.1

Cancel the common factor.

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

Step 5.1.2

Rewrite the expression.

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

Step 5.2

Divide $(x + 1) (x - 1)$ by $1$ .

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

Step 6

Cancel the common factor of $x + 1$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Divide $x - 1$ by $1$ .

$\ln (x - 1)$