Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Apply the distributive property.

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

Step 1.5

Multiply ${(x - 1)}^{2}$ by $1$ .

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

Step 2

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

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

Step 3

Factor ${(x - 1)}^{2}$ out of ${(x - 1)}^{2} x + {(x - 1)}^{2}$ .

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Step 3.1

Factor ${(x - 1)}^{2}$ out of ${(x - 1)}^{2} x$ .

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

Step 3.2

Multiply by $1$ .

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

Step 3.3

Factor ${(x - 1)}^{2}$ out of ${(x - 1)}^{2} (x) + {(x - 1)}^{2} \cdot 1$ .

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