Precalculus Examples

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

Step 1

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

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

Step 2

Simplify each term.

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Step 2.1

Simplify $2 \ln (x + 2)$ by moving $2$ inside the logarithm.

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

Step 2.2

Simplify the denominator.

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Step 2.2.1

Subtract $1$ from $2$ .

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

Step 2.2.2

Simplify.

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

Step 2.3

Cancel the common factor of $x$ .

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Step 2.3.1

Cancel the common factor.

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

Step 2.3.2

Rewrite the expression.

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

Step 2.4

The natural logarithm of $1$ is $0$ .

$\ln ({(x + 2)}^{2}) + 0$

Step 3

Add $\ln ({(x + 2)}^{2})$ and $0$ .

$\ln ({(x + 2)}^{2})$