Precalculus Examples

$2 (\ln (x) - \ln (x + 1) + \ln (x - 1))$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify $2 \ln (\frac{x (x - 1)}{x + 1})$ by moving $2$ inside the logarithm.

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

Step 5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1

Apply the product rule to $\frac{x (x - 1)}{x + 1}$ .

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

Step 5.2

Apply the product rule to $x (x - 1)$ .

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