Precalculus Examples

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

Step 1

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

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

Step 2

Cancel the common factor of $x$ .

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Step 2.1

Factor $x$ out of $x^{2}$ .

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

Step 2.2

Cancel the common factor.

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

Step 2.3

Rewrite the expression.

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

Step 3

Simplify the denominator.

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Step 3.1

Subtract $1$ from $2$ .

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

Step 3.2

Simplify.

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

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $x$ .

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Rewrite the expression.

$\log (1 (x^{2} + 1))$

Step 4.2

Multiply $x^{2} + 1$ by $1$ .

$\log (x^{2} + 1)$